{"entries":{"dist_bernoulli_cdf(DOUBLE,DOUBLE)":{"name":"dist_bernoulli_cdf","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the bernoulli distribution. 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Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_bernoulli_cdf_complement(0.5, 0);","outputTable":{"columns":[{"name":"dist_bernoulli_cdf_complement(0.5, 0)","align":"left"}],"rows":[["0.5"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_chf(DOUBLE,DOUBLE)":{"name":"dist_bernoulli_chf","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the bernoulli distribution.","examples":[{"description":"","code":"SELECT dist_bernoulli_chf(0.5, 0);","outputTable":{"columns":[{"name":"dist_bernoulli_chf(0.5, 0)","align":"left"}],"rows":[["0.6931471805599453"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_hazard(DOUBLE,DOUBLE)":{"name":"dist_bernoulli_hazard","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the bernoulli distribution.","examples":[{"description":"","code":"SELECT dist_bernoulli_hazard(0.5, 0);","outputTable":{"columns":[{"name":"dist_bernoulli_hazard(0.5, 0)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_kurtosis(DOUBLE)":{"name":"dist_bernoulli_kurtosis","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the bernoulli distribution.","examples":[{"description":"","code":"SELECT dist_bernoulli_kurtosis(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_kurtosis(0.5)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_kurtosis_excess(DOUBLE)":{"name":"dist_bernoulli_kurtosis_excess","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the bernoulli distribution.","examples":[{"description":"","code":"SELECT dist_bernoulli_kurtosis_excess(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_kurtosis_excess(0.5)","align":"left"}],"rows":[["-2.0"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_log_cdf(DOUBLE,DOUBLE)":{"name":"dist_bernoulli_log_cdf","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the bernoulli distribution. 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Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_bernoulli_log_cdf_complement(0.5, 0);","outputTable":{"columns":[{"name":"dist_bernoulli_log_cdf_complement(0.5, 0)","align":"left"}],"rows":[["-0.6931471805599453"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_log_pdf(DOUBLE,DOUBLE)":{"name":"dist_bernoulli_log_pdf","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the bernoulli distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_bernoulli_log_pdf(0.5, 1);","outputTable":{"columns":[{"name":"dist_bernoulli_log_pdf(0.5, 1)","align":"left"}],"rows":[["-0.6931471805599453"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_mean(DOUBLE)":{"name":"dist_bernoulli_mean","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the bernoulli distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_bernoulli_mean(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_mean(0.5)","align":"left"}],"rows":[["0.5"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_stddev"]},"dist_bernoulli_median(DOUBLE)":{"name":"dist_bernoulli_median","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the bernoulli distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_bernoulli_median(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_median(0.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_mode(DOUBLE)":{"name":"dist_bernoulli_mode","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the bernoulli distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_bernoulli_mode(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_mode(0.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_pdf(DOUBLE,DOUBLE)":{"name":"dist_bernoulli_pdf","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the bernoulli distribution. 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Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_bernoulli_quantile_complement(0.5, 0.05);","outputTable":{"columns":[{"name":"dist_bernoulli_quantile_complement(0.5, 0.05)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_range(DOUBLE)":{"name":"dist_bernoulli_range","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE[2]","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the bernoulli distribution.","examples":[{"description":"","code":"SELECT dist_bernoulli_range(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_range(0.5)","align":"left"}],"rows":[["(0.0, 1.0)"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_sample(DOUBLE)":{"name":"dist_bernoulli_sample","type":"scalar","categories":["Bernoulli"],"returnType":"BOOLEAN","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the bernoulli distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_bernoulli_sample(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_sample(0.5)","align":"left"}],"rows":[["true"]]}}],"relatedNames":["dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_skewness(DOUBLE)":{"name":"dist_bernoulli_skewness","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the bernoulli distribution.","examples":[{"description":"","code":"SELECT dist_bernoulli_skewness(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_skewness(0.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_stddev(DOUBLE)":{"name":"dist_bernoulli_stddev","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the standard deviation (σ) of the bernoulli distribution.","examples":[{"description":"","code":"SELECT dist_bernoulli_stddev(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_stddev(0.5)","align":"left"}],"rows":[["0.5"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean"]},"dist_bernoulli_support(DOUBLE)":{"name":"dist_bernoulli_support","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE[2]","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the bernoulli distribution.","examples":[{"description":"","code":"SELECT dist_bernoulli_support(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_support(0.5)","align":"left"}],"rows":[["(0.0, 1.0)"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_bernoulli_variance(DOUBLE)":{"name":"dist_bernoulli_variance","type":"scalar","categories":["Bernoulli"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the bernoulli distribution.","examples":[{"description":"","code":"SELECT dist_bernoulli_variance(0.5);","outputTable":{"columns":[{"name":"dist_bernoulli_variance(0.5)","align":"left"}],"rows":[["0.25"]]}}],"relatedNames":["dist_bernoulli_sample","dist_bernoulli_pdf","dist_bernoulli_cdf","dist_bernoulli_quantile","dist_bernoulli_mean","dist_bernoulli_stddev"]},"dist_beta_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_beta_cdf","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the beta distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_beta_cdf(2.0, 5.0, 0.5);","outputTable":{"columns":[{"name":"dist_beta_cdf(2.0, 5.0, 0.5)","align":"left"}],"rows":[["0.890625"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_beta_cdf_complement","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the beta distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_beta_cdf_complement(2.0, 5.0, 0.5);","outputTable":{"columns":[{"name":"dist_beta_cdf_complement(2.0, 5.0, 0.5)","align":"left"}],"rows":[["0.109375"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_beta_chf","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the beta distribution.","examples":[{"description":"","code":"SELECT dist_beta_chf(2.0, 5.0, 0.5);","outputTable":{"columns":[{"name":"dist_beta_chf(2.0, 5.0, 0.5)","align":"left"}],"rows":[["2.2129729343043585"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_beta_hazard","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the beta distribution.","examples":[{"description":"","code":"SELECT dist_beta_hazard(2.0, 5.0, 0.5);","outputTable":{"columns":[{"name":"dist_beta_hazard(2.0, 5.0, 0.5)","align":"left"}],"rows":[["8.571428571428571"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_beta_kurtosis","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the beta distribution.","examples":[{"description":"","code":"SELECT dist_beta_kurtosis(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_kurtosis(2.0, 5.0)","align":"left"}],"rows":[["2.88"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_beta_kurtosis_excess","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the beta distribution.","examples":[{"description":"","code":"SELECT dist_beta_kurtosis_excess(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_kurtosis_excess(2.0, 5.0)","align":"left"}],"rows":[["-0.12"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_beta_log_cdf","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the beta distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_beta_log_cdf(2.0, 5.0, 0.5);","outputTable":{"columns":[{"name":"dist_beta_log_cdf(2.0, 5.0, 0.5)","align":"left"}],"rows":[["-0.1158318155251217"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_beta_log_cdf_complement","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the beta distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_beta_log_cdf_complement(2.0, 5.0, 0.5);","outputTable":{"columns":[{"name":"dist_beta_log_cdf_complement(2.0, 5.0, 0.5)","align":"left"}],"rows":[["-2.2129729343043585"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_beta_log_pdf","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the beta distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_beta_log_pdf(2.0, 5.0, 0.5);","outputTable":{"columns":[{"name":"dist_beta_log_pdf(2.0, 5.0, 0.5)","align":"left"}],"rows":[["-0.06453852113757129"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_mean(DOUBLE,DOUBLE)":{"name":"dist_beta_mean","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the beta distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_beta_mean(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_mean(2.0, 5.0)","align":"left"}],"rows":[["0.2857142857142857"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_stddev"]},"dist_beta_median(DOUBLE,DOUBLE)":{"name":"dist_beta_median","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the beta distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_beta_median(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_median(2.0, 5.0)","align":"left"}],"rows":[["0.26444998329566005"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_mode(DOUBLE,DOUBLE)":{"name":"dist_beta_mode","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the beta distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_beta_mode(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_mode(2.0, 5.0)","align":"left"}],"rows":[["0.2"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_beta_pdf","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the beta distribution. Returns the probability densityat point x for a beta distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_beta_pdf(2.0, 5.0, 0.5);","outputTable":{"columns":[{"name":"dist_beta_pdf(2.0, 5.0, 0.5)","align":"left"}],"rows":[["0.9374999999999999"]]}}],"relatedNames":["dist_beta_sample","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_beta_quantile","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the beta distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_beta_quantile(2.0, 5.0, 0.95);","outputTable":{"columns":[{"name":"dist_beta_quantile(2.0, 5.0, 0.95)","align":"left"}],"rows":[["0.5818034092520259"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_mean","dist_beta_stddev"]},"dist_beta_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_beta_quantile_complement","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the beta distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_beta_quantile_complement(2.0, 5.0, 0.05);","outputTable":{"columns":[{"name":"dist_beta_quantile_complement(2.0, 5.0, 0.05)","align":"left"}],"rows":[["0.5818034092520259"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_range(DOUBLE,DOUBLE)":{"name":"dist_beta_range","type":"scalar","categories":["Beta"],"returnType":"DOUBLE[2]","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the beta distribution.","examples":[{"description":"","code":"SELECT dist_beta_range(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_range(2.0, 5.0)","align":"left"}],"rows":[["(0.0, 1.0)"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_sample(DOUBLE,DOUBLE)":{"name":"dist_beta_sample","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the beta distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_beta_sample(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_sample(2.0, 5.0)","align":"left"}],"rows":[["0.49894986644503236"]]}}],"relatedNames":["dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_skewness(DOUBLE,DOUBLE)":{"name":"dist_beta_skewness","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the beta distribution.","examples":[{"description":"","code":"SELECT dist_beta_skewness(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_skewness(2.0, 5.0)","align":"left"}],"rows":[["0.5962847939999439"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_stddev(DOUBLE,DOUBLE)":{"name":"dist_beta_stddev","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the standard deviation (σ) of the beta distribution.","examples":[{"description":"","code":"SELECT dist_beta_stddev(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_stddev(2.0, 5.0)","align":"left"}],"rows":[["0.15971914124998499"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean"]},"dist_beta_support(DOUBLE,DOUBLE)":{"name":"dist_beta_support","type":"scalar","categories":["Beta"],"returnType":"DOUBLE[2]","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the beta distribution.","examples":[{"description":"","code":"SELECT dist_beta_support(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_support(2.0, 5.0)","align":"left"}],"rows":[["(0.0, 1.0)"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_beta_variance(DOUBLE,DOUBLE)":{"name":"dist_beta_variance","type":"scalar","categories":["Beta"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the beta distribution.","examples":[{"description":"","code":"SELECT dist_beta_variance(2.0, 5.0);","outputTable":{"columns":[{"name":"dist_beta_variance(2.0, 5.0)","align":"left"}],"rows":[["0.025510204081632654"]]}}],"relatedNames":["dist_beta_sample","dist_beta_pdf","dist_beta_cdf","dist_beta_quantile","dist_beta_mean","dist_beta_stddev"]},"dist_binomial_cdf(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_binomial_cdf","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the binomial distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_binomial_cdf(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_binomial_cdf(10, 0.5, 5)","align":"left"}],"rows":[["0.623046875"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_quantile"]},"dist_binomial_cdf_complement(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_binomial_cdf_complement","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the binomial distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_binomial_cdf_complement(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_binomial_cdf_complement(10, 0.5, 5)","align":"left"}],"rows":[["0.376953125"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_chf(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_binomial_chf","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the binomial distribution.","examples":[{"description":"","code":"SELECT dist_binomial_chf(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_binomial_chf(10, 0.5, 5)","align":"left"}],"rows":[["0.9756344361346222"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_hazard(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_binomial_hazard","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the binomial distribution.","examples":[{"description":"","code":"SELECT dist_binomial_hazard(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_binomial_hazard(10, 0.5, 5)","align":"left"}],"rows":[["0.652849740932643"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_kurtosis(BIGINT,DOUBLE)":{"name":"dist_binomial_kurtosis","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the binomial distribution.","examples":[{"description":"","code":"SELECT dist_binomial_kurtosis(10, 0.5);","outputTable":{"columns":[{"name":"dist_binomial_kurtosis(10, 0.5)","align":"left"}],"rows":[["2.8"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_kurtosis_excess(BIGINT,DOUBLE)":{"name":"dist_binomial_kurtosis_excess","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the binomial distribution.","examples":[{"description":"","code":"SELECT dist_binomial_kurtosis_excess(10, 0.5);","outputTable":{"columns":[{"name":"dist_binomial_kurtosis_excess(10, 0.5)","align":"left"}],"rows":[["-0.2"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_log_cdf(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_binomial_log_cdf","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the binomial distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_binomial_log_cdf(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_binomial_log_cdf(10, 0.5, 5)","align":"left"}],"rows":[["-0.4731335222546632"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_log_cdf_complement(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_binomial_log_cdf_complement","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the binomial distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_binomial_log_cdf_complement(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_binomial_log_cdf_complement(10, 0.5, 5)","align":"left"}],"rows":[["-0.9756344361346222"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_log_pdf(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_binomial_log_pdf","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the binomial distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_binomial_log_pdf(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_binomial_log_pdf(10, 0.5, 5)","align":"left"}],"rows":[["-1.402042718088029"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_median(BIGINT,DOUBLE)":{"name":"dist_binomial_median","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the binomial distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_binomial_median(10, 0.5);","outputTable":{"columns":[{"name":"dist_binomial_median(10, 0.5)","align":"left"}],"rows":[["5.0"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_mode(BIGINT,DOUBLE)":{"name":"dist_binomial_mode","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the binomial distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_binomial_mode(10, 0.5);","outputTable":{"columns":[{"name":"dist_binomial_mode(10, 0.5)","align":"left"}],"rows":[["5.0"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_pdf(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_binomial_pdf","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the binomial distribution. Returns the probability densityat point x for a binomial distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_binomial_pdf(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_binomial_pdf(10, 0.5, 5)","align":"left"}],"rows":[["0.2460937500000002"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_quantile(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_binomial_quantile","type":"scalar","categories":["Binomial"],"returnType":"BIGINT","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the binomial distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_binomial_quantile(10, 0.5, 0.95);","outputTable":{"columns":[{"name":"dist_binomial_quantile(10, 0.5, 0.95)","align":"left"}],"rows":[["8"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf"]},"dist_binomial_quantile_complement(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_binomial_quantile_complement","type":"scalar","categories":["Binomial"],"returnType":"BIGINT","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the binomial distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_binomial_quantile_complement(10, 0.5, 0.05);","outputTable":{"columns":[{"name":"dist_binomial_quantile_complement(10, 0.5, 0.05)","align":"left"}],"rows":[["8"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_range(BIGINT,DOUBLE)":{"name":"dist_binomial_range","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE[2]","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the binomial distribution.","examples":[{"description":"","code":"SELECT dist_binomial_range(10, 0.5);","outputTable":{"columns":[{"name":"dist_binomial_range(10, 0.5)","align":"left"}],"rows":[["(0.0, 10.0)"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_sample(BIGINT,DOUBLE)":{"name":"dist_binomial_sample","type":"scalar","categories":["Binomial"],"returnType":"BIGINT","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the binomial distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_binomial_sample(10, 0.5);","outputTable":{"columns":[{"name":"dist_binomial_sample(10, 0.5)","align":"left"}],"rows":[["2"]]}}],"relatedNames":["dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_skewness(BIGINT,DOUBLE)":{"name":"dist_binomial_skewness","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the binomial distribution.","examples":[{"description":"","code":"SELECT dist_binomial_skewness(10, 0.5);","outputTable":{"columns":[{"name":"dist_binomial_skewness(10, 0.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_support(BIGINT,DOUBLE)":{"name":"dist_binomial_support","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE[2]","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the binomial distribution.","examples":[{"description":"","code":"SELECT dist_binomial_support(10, 0.5);","outputTable":{"columns":[{"name":"dist_binomial_support(10, 0.5)","align":"left"}],"rows":[["(0.0, 10.0)"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_binomial_variance(BIGINT,DOUBLE)":{"name":"dist_binomial_variance","type":"scalar","categories":["Binomial"],"returnType":"DOUBLE","parameters":[{"name":"trials","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the binomial distribution.","examples":[{"description":"","code":"SELECT dist_binomial_variance(10, 0.5);","outputTable":{"columns":[{"name":"dist_binomial_variance(10, 0.5)","align":"left"}],"rows":[["2.5"]]}}],"relatedNames":["dist_binomial_sample","dist_binomial_pdf","dist_binomial_cdf","dist_binomial_quantile"]},"dist_cauchy_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_cauchy_cdf","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the cauchy distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_cauchy_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_cauchy_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.6475836176504333"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_quantile"]},"dist_cauchy_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_cauchy_cdf_complement","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the cauchy distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_cauchy_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_cauchy_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["0.35241638234956674"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_cauchy_chf","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the cauchy distribution.","examples":[{"description":"","code":"SELECT dist_cauchy_chf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_cauchy_chf(0, 1.0, 0.5)","align":"left"}],"rows":[["1.0429418980620317"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_cauchy_hazard","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the cauchy distribution.","examples":[{"description":"","code":"SELECT dist_cauchy_hazard(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_cauchy_hazard(0, 1.0, 0.5)","align":"left"}],"rows":[["0.7225768202070803"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_cauchy_log_cdf","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the cauchy distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_cauchy_log_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_cauchy_log_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.4345073545177282"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_cauchy_log_cdf_complement","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the cauchy distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_cauchy_log_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_cauchy_log_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["-1.0429418980620317"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_cauchy_log_pdf","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the cauchy distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_cauchy_log_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_cauchy_log_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-1.3678734371636099"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_median(DOUBLE,DOUBLE)":{"name":"dist_cauchy_median","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the cauchy distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_cauchy_median(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_cauchy_median(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_mode(DOUBLE,DOUBLE)":{"name":"dist_cauchy_mode","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the cauchy distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_cauchy_mode(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_cauchy_mode(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_cauchy_pdf","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the cauchy distribution. Returns the probability densityat point x for a cauchy distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_cauchy_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_cauchy_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.25464790894703254"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_cauchy_quantile","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the cauchy distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_cauchy_quantile(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_cauchy_quantile(0, 1.0, 0.95)","align":"left"}],"rows":[["6.313751514675038"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf"]},"dist_cauchy_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_cauchy_quantile_complement","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the cauchy distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_cauchy_quantile_complement(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_cauchy_quantile_complement(0, 1.0, 0.95)","align":"left"}],"rows":[["-6.313751514675038"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_range(DOUBLE,DOUBLE)":{"name":"dist_cauchy_range","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE[2]","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the cauchy distribution.","examples":[{"description":"","code":"SELECT dist_cauchy_range(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_cauchy_range(0.0, 1.0)","align":"left"}],"rows":[["(-inf, inf)"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_sample(DOUBLE,DOUBLE)":{"name":"dist_cauchy_sample","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the cauchy distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_cauchy_sample(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_cauchy_sample(0.0, 1.0)","align":"left"}],"rows":[["-1.2266264270965836"]]}}],"relatedNames":["dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_cauchy_support(DOUBLE,DOUBLE)":{"name":"dist_cauchy_support","type":"scalar","categories":["Cauchy"],"returnType":"DOUBLE[2]","parameters":[{"name":"x","type":"DOUBLE","paramType":"positional","description":""},{"name":"y","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the cauchy distribution.","examples":[{"description":"","code":"SELECT dist_cauchy_support(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_cauchy_support(0.0, 1.0)","align":"left"}],"rows":[["(-inf, inf)"]]}}],"relatedNames":["dist_cauchy_sample","dist_cauchy_pdf","dist_cauchy_cdf","dist_cauchy_quantile"]},"dist_chi_squared_cdf(DOUBLE,DOUBLE)":{"name":"dist_chi_squared_cdf","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the chi_squared distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_chi_squared_cdf(5, 3.0);","outputTable":{"columns":[{"name":"dist_chi_squared_cdf(5, 3.0)","align":"left"}],"rows":[["0.3000141641213725"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_cdf_complement(DOUBLE,DOUBLE)":{"name":"dist_chi_squared_cdf_complement","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the chi_squared distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_chi_squared_cdf_complement(5, 3.0);","outputTable":{"columns":[{"name":"dist_chi_squared_cdf_complement(5, 3.0)","align":"left"}],"rows":[["0.6999858358786275"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_chf(DOUBLE,DOUBLE)":{"name":"dist_chi_squared_chf","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the chi_squared distribution.","examples":[{"description":"","code":"SELECT dist_chi_squared_chf(5, 3.0);","outputTable":{"columns":[{"name":"dist_chi_squared_chf(5, 3.0)","align":"left"}],"rows":[["0.3566951786025554"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_hazard(DOUBLE,DOUBLE)":{"name":"dist_chi_squared_hazard","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the chi_squared distribution.","examples":[{"description":"","code":"SELECT dist_chi_squared_hazard(5, 3.0);","outputTable":{"columns":[{"name":"dist_chi_squared_hazard(5, 3.0)","align":"left"}],"rows":[["0.2202620708892502"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_kurtosis(DOUBLE)":{"name":"dist_chi_squared_kurtosis","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the chi_squared distribution.","examples":[{"description":"","code":"SELECT dist_chi_squared_kurtosis(5);","outputTable":{"columns":[{"name":"dist_chi_squared_kurtosis(5)","align":"left"}],"rows":[["5.4"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_kurtosis_excess(DOUBLE)":{"name":"dist_chi_squared_kurtosis_excess","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the chi_squared distribution.","examples":[{"description":"","code":"SELECT dist_chi_squared_kurtosis_excess(5);","outputTable":{"columns":[{"name":"dist_chi_squared_kurtosis_excess(5)","align":"left"}],"rows":[["2.4"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_log_cdf(DOUBLE,DOUBLE)":{"name":"dist_chi_squared_log_cdf","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the chi_squared distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_chi_squared_log_cdf(5, 3.0);","outputTable":{"columns":[{"name":"dist_chi_squared_log_cdf(5, 3.0)","align":"left"}],"rows":[["-1.203925591702561"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_log_cdf_complement(DOUBLE,DOUBLE)":{"name":"dist_chi_squared_log_cdf_complement","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the chi_squared distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_chi_squared_log_cdf_complement(5, 3.0);","outputTable":{"columns":[{"name":"dist_chi_squared_log_cdf_complement(5, 3.0)","align":"left"}],"rows":[["-0.3566951786025554"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_log_pdf(DOUBLE,DOUBLE)":{"name":"dist_chi_squared_log_pdf","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the chi_squared distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_chi_squared_log_pdf(5, 3.0);","outputTable":{"columns":[{"name":"dist_chi_squared_log_pdf(5, 3.0)","align":"left"}],"rows":[["-1.8696323888706177"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_mean(DOUBLE)":{"name":"dist_chi_squared_mean","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the chi_squared distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_chi_squared_mean(5);","outputTable":{"columns":[{"name":"dist_chi_squared_mean(5)","align":"left"}],"rows":[["5.0"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_stddev"]},"dist_chi_squared_median(DOUBLE)":{"name":"dist_chi_squared_median","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the chi_squared distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_chi_squared_median(5);","outputTable":{"columns":[{"name":"dist_chi_squared_median(5)","align":"left"}],"rows":[["4.351460191095527"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_mode(DOUBLE)":{"name":"dist_chi_squared_mode","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the chi_squared distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_chi_squared_mode(5);","outputTable":{"columns":[{"name":"dist_chi_squared_mode(5)","align":"left"}],"rows":[["3.0"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_pdf(DOUBLE,DOUBLE)":{"name":"dist_chi_squared_pdf","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the chi_squared distribution. Returns the probability densityat point x for a chi_squared distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_chi_squared_pdf(5, 3.0);","outputTable":{"columns":[{"name":"dist_chi_squared_pdf(5, 3.0)","align":"left"}],"rows":[["0.1541803298037693"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_cdf","dist_chi_squared_quantile","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_quantile(DOUBLE,DOUBLE)":{"name":"dist_chi_squared_quantile","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the chi_squared distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_chi_squared_quantile(5, 0.95);","outputTable":{"columns":[{"name":"dist_chi_squared_quantile(5, 0.95)","align":"left"}],"rows":[["11.070497693516351"]]}}],"relatedNames":["dist_chi_squared_sample","dist_chi_squared_pdf","dist_chi_squared_cdf","dist_chi_squared_mean","dist_chi_squared_stddev"]},"dist_chi_squared_quantile_complement(DOUBLE,DOUBLE)":{"name":"dist_chi_squared_quantile_complement","type":"scalar","categories":["Chi-squared"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the chi_squared distribution. 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Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_exponential_log_cdf_complement(1.0, 0.5);","outputTable":{"columns":[{"name":"dist_exponential_log_cdf_complement(1.0, 0.5)","align":"left"}],"rows":[["-0.5"]]}}],"relatedNames":["dist_exponential_sample","dist_exponential_pdf","dist_exponential_cdf","dist_exponential_quantile","dist_exponential_mean","dist_exponential_stddev"]},"dist_exponential_log_pdf(DOUBLE,DOUBLE)":{"name":"dist_exponential_log_pdf","type":"scalar","categories":["Exponential"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the exponential distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_exponential_log_pdf(1.0, 0.5);","outputTable":{"columns":[{"name":"dist_exponential_log_pdf(1.0, 0.5)","align":"left"}],"rows":[["-0.5"]]}}],"relatedNames":["dist_exponential_sample","dist_exponential_pdf","dist_exponential_cdf","dist_exponential_quantile","dist_exponential_mean","dist_exponential_stddev"]},"dist_exponential_mean(DOUBLE)":{"name":"dist_exponential_mean","type":"scalar","categories":["Exponential"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the exponential distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_exponential_mean(1.0);","outputTable":{"columns":[{"name":"dist_exponential_mean(1.0)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_exponential_sample","dist_exponential_pdf","dist_exponential_cdf","dist_exponential_quantile","dist_exponential_stddev"]},"dist_exponential_median(DOUBLE)":{"name":"dist_exponential_median","type":"scalar","categories":["Exponential"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the exponential distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_exponential_median(1.0);","outputTable":{"columns":[{"name":"dist_exponential_median(1.0)","align":"left"}],"rows":[["0.6931471805599453"]]}}],"relatedNames":["dist_exponential_sample","dist_exponential_pdf","dist_exponential_cdf","dist_exponential_quantile","dist_exponential_mean","dist_exponential_stddev"]},"dist_exponential_mode(DOUBLE)":{"name":"dist_exponential_mode","type":"scalar","categories":["Exponential"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the exponential distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_exponential_mode(1.0);","outputTable":{"columns":[{"name":"dist_exponential_mode(1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_exponential_sample","dist_exponential_pdf","dist_exponential_cdf","dist_exponential_quantile","dist_exponential_mean","dist_exponential_stddev"]},"dist_exponential_pdf(DOUBLE,DOUBLE)":{"name":"dist_exponential_pdf","type":"scalar","categories":["Exponential"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the exponential distribution. 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Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_extreme_value_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_extreme_value_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.545239211892605"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_quantile"]},"dist_extreme_value_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_extreme_value_cdf_complement","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the extreme_value distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_extreme_value_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_extreme_value_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["0.45476078810739495"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_extreme_value_chf","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the extreme_value distribution.","examples":[{"description":"","code":"SELECT dist_extreme_value_chf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_extreme_value_chf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.7879837387044486"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_extreme_value_hazard","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the extreme_value distribution.","examples":[{"description":"","code":"SELECT dist_extreme_value_hazard(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_extreme_value_hazard(0, 1.0, 0.5)","align":"left"}],"rows":[["0.7272049559653766"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_extreme_value_kurtosis","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the extreme_value distribution.","examples":[{"description":"","code":"SELECT dist_extreme_value_kurtosis(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_extreme_value_kurtosis(0.0, 1.0)","align":"left"}],"rows":[["5.4"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_extreme_value_kurtosis_excess","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the extreme_value distribution.","examples":[{"description":"","code":"SELECT dist_extreme_value_kurtosis_excess(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_extreme_value_kurtosis_excess(0.0, 1.0)","align":"left"}],"rows":[["2.4"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_extreme_value_log_cdf","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the extreme_value distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_extreme_value_log_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_extreme_value_log_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.6065306597126334"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_extreme_value_log_cdf_complement","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the extreme_value distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_extreme_value_log_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_extreme_value_log_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.7879837387044486"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_extreme_value_log_pdf","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the extreme_value distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_extreme_value_log_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_extreme_value_log_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-1.1065306597126334"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_median(DOUBLE,DOUBLE)":{"name":"dist_extreme_value_median","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the extreme_value distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_extreme_value_median(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_extreme_value_median(0.0, 1.0)","align":"left"}],"rows":[["0.36651292058166435"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_mode(DOUBLE,DOUBLE)":{"name":"dist_extreme_value_mode","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the extreme_value distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_extreme_value_mode(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_extreme_value_mode(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_extreme_value_pdf","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the extreme_value distribution. Returns the probability densityat point x for a extreme_value distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_extreme_value_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_extreme_value_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.3307042988904181"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_extreme_value_quantile","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the extreme_value distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_extreme_value_quantile(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_extreme_value_quantile(0, 1.0, 0.95)","align":"left"}],"rows":[["2.9701952490421637"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf"]},"dist_extreme_value_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_extreme_value_quantile_complement","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the extreme_value distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_extreme_value_quantile_complement(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_extreme_value_quantile_complement(0, 1.0, 0.95)","align":"left"}],"rows":[["-1.0971887003649483"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_range(DOUBLE,DOUBLE)":{"name":"dist_extreme_value_range","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE[2]","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the extreme_value distribution.","examples":[{"description":"","code":"SELECT dist_extreme_value_range(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_extreme_value_range(0.0, 1.0)","align":"left"}],"rows":[["(-inf, inf)"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_sample(DOUBLE,DOUBLE)":{"name":"dist_extreme_value_sample","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the extreme_value distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_extreme_value_sample(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_extreme_value_sample(0.0, 1.0)","align":"left"}],"rows":[["1.02415581208542"]]}}],"relatedNames":["dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_skewness(DOUBLE,DOUBLE)":{"name":"dist_extreme_value_skewness","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the extreme_value distribution.","examples":[{"description":"","code":"SELECT dist_extreme_value_skewness(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_extreme_value_skewness(0.0, 1.0)","align":"left"}],"rows":[["1.1395470994046486"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_support(DOUBLE,DOUBLE)":{"name":"dist_extreme_value_support","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE[2]","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the extreme_value distribution.","examples":[{"description":"","code":"SELECT dist_extreme_value_support(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_extreme_value_support(0.0, 1.0)","align":"left"}],"rows":[["(-1.7976931348623157e+308, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_extreme_value_variance(DOUBLE,DOUBLE)":{"name":"dist_extreme_value_variance","type":"scalar","categories":["Extreme Value"],"returnType":"DOUBLE","parameters":[{"name":"real","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the extreme_value distribution.","examples":[{"description":"","code":"SELECT dist_extreme_value_variance(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_extreme_value_variance(0.0, 1.0)","align":"left"}],"rows":[["1.6449340668482264"]]}}],"relatedNames":["dist_extreme_value_sample","dist_extreme_value_pdf","dist_extreme_value_cdf","dist_extreme_value_quantile"]},"dist_fisher_f_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_fisher_f_cdf","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the fisher_f distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_fisher_f_cdf(5, 10, 0.5);","outputTable":{"columns":[{"name":"dist_fisher_f_cdf(5, 10, 0.5)","align":"left"}],"rows":[["0.22997511934989848"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_quantile"]},"dist_fisher_f_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_fisher_f_cdf_complement","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the fisher_f distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_fisher_f_cdf_complement(5, 10, 0.5);","outputTable":{"columns":[{"name":"dist_fisher_f_cdf_complement(5, 10, 0.5)","align":"left"}],"rows":[["0.7700248806501016"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_fisher_f_chf","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the fisher_f distribution.","examples":[{"description":"","code":"SELECT dist_fisher_f_chf(5, 10, 0.5);","outputTable":{"columns":[{"name":"dist_fisher_f_chf(5, 10, 0.5)","align":"left"}],"rows":[["0.26133245212384676"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_fisher_f_hazard","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the fisher_f distribution.","examples":[{"description":"","code":"SELECT dist_fisher_f_hazard(5, 10, 0.5);","outputTable":{"columns":[{"name":"dist_fisher_f_hazard(5, 10, 0.5)","align":"left"}],"rows":[["0.8929672534608285"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_fisher_f_kurtosis","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the fisher_f distribution.","examples":[{"description":"","code":"SELECT dist_fisher_f_kurtosis(5, 10);","outputTable":{"columns":[{"name":"dist_fisher_f_kurtosis(5, 10)","align":"left"}],"rows":[["53.86153846153846"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_fisher_f_kurtosis_excess","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the fisher_f distribution.","examples":[{"description":"","code":"SELECT dist_fisher_f_kurtosis_excess(5, 10);","outputTable":{"columns":[{"name":"dist_fisher_f_kurtosis_excess(5, 10)","align":"left"}],"rows":[["50.86153846153846"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_fisher_f_log_cdf","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the fisher_f distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_fisher_f_log_cdf(5, 10, 0.5);","outputTable":{"columns":[{"name":"dist_fisher_f_log_cdf(5, 10, 0.5)","align":"left"}],"rows":[["-1.469784152650039"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_fisher_f_log_cdf_complement","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the fisher_f distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_fisher_f_log_cdf_complement(5, 10, 0.5);","outputTable":{"columns":[{"name":"dist_fisher_f_log_cdf_complement(5, 10, 0.5)","align":"left"}],"rows":[["-0.26133245212384676"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_fisher_f_log_pdf","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the fisher_f distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_fisher_f_log_pdf(5, 10, 0.5);","outputTable":{"columns":[{"name":"dist_fisher_f_log_pdf(5, 10, 0.5)","align":"left"}],"rows":[["-0.374537821158486"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_median(DOUBLE,DOUBLE)":{"name":"dist_fisher_f_median","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the fisher_f distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_fisher_f_median(5, 10);","outputTable":{"columns":[{"name":"dist_fisher_f_median(5, 10)","align":"left"}],"rows":[["0.931933160851048"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_mode(DOUBLE,DOUBLE)":{"name":"dist_fisher_f_mode","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the fisher_f distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_fisher_f_mode(5, 10);","outputTable":{"columns":[{"name":"dist_fisher_f_mode(5, 10)","align":"left"}],"rows":[["0.5"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_fisher_f_pdf","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the fisher_f distribution. Returns the probability densityat point x for a fisher_f distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_fisher_f_pdf(5, 10, 0.5);","outputTable":{"columns":[{"name":"dist_fisher_f_pdf(5, 10, 0.5)","align":"left"}],"rows":[["0.6876070027706235"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_cdf","dist_fisher_f_quantile"]},"dist_fisher_f_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_fisher_f_quantile","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the fisher_f distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_fisher_f_quantile(5, 10, 0.95);","outputTable":{"columns":[{"name":"dist_fisher_f_quantile(5, 10, 0.95)","align":"left"}],"rows":[["3.3258345304130104"]]}}],"relatedNames":["dist_fisher_f_sample","dist_fisher_f_pdf","dist_fisher_f_cdf"]},"dist_fisher_f_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_fisher_f_quantile_complement","type":"scalar","categories":["Fisher F"],"returnType":"DOUBLE","parameters":[{"name":"d1","type":"DOUBLE","paramType":"positional","description":""},{"name":"d2","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the fisher_f distribution. 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Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_gamma_cdf(2.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_gamma_cdf(2.0, 1.0, 1.5)","align":"left"}],"rows":[["0.44217459962892547"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_gamma_cdf_complement","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the gamma distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_gamma_cdf_complement(2.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_gamma_cdf_complement(2.0, 1.0, 1.5)","align":"left"}],"rows":[["0.5578254003710745"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_gamma_chf","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the gamma distribution.","examples":[{"description":"","code":"SELECT dist_gamma_chf(2.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_gamma_chf(2.0, 1.0, 1.5)","align":"left"}],"rows":[["0.583709268125845"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_gamma_hazard","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the gamma distribution.","examples":[{"description":"","code":"SELECT dist_gamma_hazard(2.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_gamma_hazard(2.0, 1.0, 1.5)","align":"left"}],"rows":[["0.6"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_gamma_kurtosis","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the gamma distribution.","examples":[{"description":"","code":"SELECT dist_gamma_kurtosis(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_kurtosis(2.0, 1.0)","align":"left"}],"rows":[["6.0"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_gamma_kurtosis_excess","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the gamma distribution.","examples":[{"description":"","code":"SELECT dist_gamma_kurtosis_excess(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_kurtosis_excess(2.0, 1.0)","align":"left"}],"rows":[["3.0"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_gamma_log_cdf","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the gamma distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_gamma_log_cdf(2.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_gamma_log_cdf(2.0, 1.0, 1.5)","align":"left"}],"rows":[["-0.8160504531201044"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_gamma_log_cdf_complement","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the gamma distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_gamma_log_cdf_complement(2.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_gamma_log_cdf_complement(2.0, 1.0, 1.5)","align":"left"}],"rows":[["-0.583709268125845"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_gamma_log_pdf","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the gamma distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_gamma_log_pdf(2.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_gamma_log_pdf(2.0, 1.0, 1.5)","align":"left"}],"rows":[["-1.0945348918918356"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_mean(DOUBLE,DOUBLE)":{"name":"dist_gamma_mean","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the gamma distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_gamma_mean(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_mean(2.0, 1.0)","align":"left"}],"rows":[["2.0"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_stddev"]},"dist_gamma_median(DOUBLE,DOUBLE)":{"name":"dist_gamma_median","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the gamma distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_gamma_median(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_median(2.0, 1.0)","align":"left"}],"rows":[["1.6783469900166605"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_mode(DOUBLE,DOUBLE)":{"name":"dist_gamma_mode","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the gamma distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_gamma_mode(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_mode(2.0, 1.0)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_gamma_pdf","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the gamma distribution. Returns the probability densityat point x for a gamma distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_gamma_pdf(2.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_gamma_pdf(2.0, 1.0, 1.5)","align":"left"}],"rows":[["0.3346952402226447"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_gamma_quantile","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the gamma distribution. 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Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_gamma_quantile_complement(2.0, 1.0, 0.05);","outputTable":{"columns":[{"name":"dist_gamma_quantile_complement(2.0, 1.0, 0.05)","align":"left"}],"rows":[["4.743864518390578"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_range(DOUBLE,DOUBLE)":{"name":"dist_gamma_range","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE[2]","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the gamma distribution.","examples":[{"description":"","code":"SELECT dist_gamma_range(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_range(2.0, 1.0)","align":"left"}],"rows":[["(0.0, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_sample(DOUBLE,DOUBLE)":{"name":"dist_gamma_sample","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the gamma distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_gamma_sample(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_sample(2.0, 1.0)","align":"left"}],"rows":[["1.036968725054842"]]}}],"relatedNames":["dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_skewness(DOUBLE,DOUBLE)":{"name":"dist_gamma_skewness","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the gamma distribution.","examples":[{"description":"","code":"SELECT dist_gamma_skewness(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_skewness(2.0, 1.0)","align":"left"}],"rows":[["1.414213562373095"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_stddev(DOUBLE,DOUBLE)":{"name":"dist_gamma_stddev","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the standard deviation (σ) of the gamma distribution.","examples":[{"description":"","code":"SELECT dist_gamma_stddev(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_stddev(2.0, 1.0)","align":"left"}],"rows":[["1.4142135623730951"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean"]},"dist_gamma_support(DOUBLE,DOUBLE)":{"name":"dist_gamma_support","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE[2]","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the gamma distribution.","examples":[{"description":"","code":"SELECT dist_gamma_support(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_support(2.0, 1.0)","align":"left"}],"rows":[["(2.2250738585072014e-308, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_gamma_variance(DOUBLE,DOUBLE)":{"name":"dist_gamma_variance","type":"scalar","categories":["Gamma"],"returnType":"DOUBLE","parameters":[{"name":"alpha","type":"DOUBLE","paramType":"positional","description":""},{"name":"beta","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the gamma distribution.","examples":[{"description":"","code":"SELECT dist_gamma_variance(2.0, 1.0);","outputTable":{"columns":[{"name":"dist_gamma_variance(2.0, 1.0)","align":"left"}],"rows":[["2.0"]]}}],"relatedNames":["dist_gamma_sample","dist_gamma_pdf","dist_gamma_cdf","dist_gamma_quantile","dist_gamma_mean","dist_gamma_stddev"]},"dist_geometric_cdf(DOUBLE,DOUBLE)":{"name":"dist_geometric_cdf","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the geometric distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_geometric_cdf(0.5, 2);","outputTable":{"columns":[{"name":"dist_geometric_cdf(0.5, 2)","align":"left"}],"rows":[["0.875"]]}}],"relatedNames":["dist_geometric_sample","dist_geometric_pdf","dist_geometric_quantile","dist_geometric_mean","dist_geometric_stddev"]},"dist_geometric_cdf_complement(DOUBLE,DOUBLE)":{"name":"dist_geometric_cdf_complement","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the geometric distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_geometric_cdf_complement(0.5, 2);","outputTable":{"columns":[{"name":"dist_geometric_cdf_complement(0.5, 2)","align":"left"}],"rows":[["0.12500000000000003"]]}}],"relatedNames":["dist_geometric_sample","dist_geometric_pdf","dist_geometric_cdf","dist_geometric_quantile","dist_geometric_mean","dist_geometric_stddev"]},"dist_geometric_chf(DOUBLE,DOUBLE)":{"name":"dist_geometric_chf","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the geometric distribution.","examples":[{"description":"","code":"SELECT dist_geometric_chf(0.5, 2);","outputTable":{"columns":[{"name":"dist_geometric_chf(0.5, 2)","align":"left"}],"rows":[["2.0794415416798357"]]}}],"relatedNames":["dist_geometric_sample","dist_geometric_pdf","dist_geometric_cdf","dist_geometric_quantile","dist_geometric_mean","dist_geometric_stddev"]},"dist_geometric_hazard(DOUBLE,DOUBLE)":{"name":"dist_geometric_hazard","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the geometric distribution.","examples":[{"description":"","code":"SELECT dist_geometric_hazard(0.5, 2);","outputTable":{"columns":[{"name":"dist_geometric_hazard(0.5, 2)","align":"left"}],"rows":[["0.9999999999999998"]]}}],"relatedNames":["dist_geometric_sample","dist_geometric_pdf","dist_geometric_cdf","dist_geometric_quantile","dist_geometric_mean","dist_geometric_stddev"]},"dist_geometric_kurtosis(DOUBLE)":{"name":"dist_geometric_kurtosis","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the geometric distribution.","examples":[{"description":"","code":"SELECT dist_geometric_kurtosis(0.5);","outputTable":{"columns":[{"name":"dist_geometric_kurtosis(0.5)","align":"left"}],"rows":[["9.5"]]}}],"relatedNames":["dist_geometric_sample","dist_geometric_pdf","dist_geometric_cdf","dist_geometric_quantile","dist_geometric_mean","dist_geometric_stddev"]},"dist_geometric_kurtosis_excess(DOUBLE)":{"name":"dist_geometric_kurtosis_excess","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the geometric distribution.","examples":[{"description":"","code":"SELECT dist_geometric_kurtosis_excess(0.5);","outputTable":{"columns":[{"name":"dist_geometric_kurtosis_excess(0.5)","align":"left"}],"rows":[["6.5"]]}}],"relatedNames":["dist_geometric_sample","dist_geometric_pdf","dist_geometric_cdf","dist_geometric_quantile","dist_geometric_mean","dist_geometric_stddev"]},"dist_geometric_log_cdf(DOUBLE,DOUBLE)":{"name":"dist_geometric_log_cdf","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the geometric distribution. 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Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_geometric_log_pdf(0.5, 2);","outputTable":{"columns":[{"name":"dist_geometric_log_pdf(0.5, 2)","align":"left"}],"rows":[["-2.0794415416798357"]]}}],"relatedNames":["dist_geometric_sample","dist_geometric_pdf","dist_geometric_cdf","dist_geometric_quantile","dist_geometric_mean","dist_geometric_stddev"]},"dist_geometric_mean(DOUBLE)":{"name":"dist_geometric_mean","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the geometric distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_geometric_mean(0.5);","outputTable":{"columns":[{"name":"dist_geometric_mean(0.5)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_geometric_sample","dist_geometric_pdf","dist_geometric_cdf","dist_geometric_quantile","dist_geometric_stddev"]},"dist_geometric_median(DOUBLE)":{"name":"dist_geometric_median","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the geometric distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_geometric_median(0.5);","outputTable":{"columns":[{"name":"dist_geometric_median(0.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_geometric_sample","dist_geometric_pdf","dist_geometric_cdf","dist_geometric_quantile","dist_geometric_mean","dist_geometric_stddev"]},"dist_geometric_mode(DOUBLE)":{"name":"dist_geometric_mode","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the geometric distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_geometric_mode(0.5);","outputTable":{"columns":[{"name":"dist_geometric_mode(0.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_geometric_sample","dist_geometric_pdf","dist_geometric_cdf","dist_geometric_quantile","dist_geometric_mean","dist_geometric_stddev"]},"dist_geometric_pdf(DOUBLE,DOUBLE)":{"name":"dist_geometric_pdf","type":"scalar","categories":["Geometric"],"returnType":"DOUBLE","parameters":[{"name":"p","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the geometric distribution. 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Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_laplace_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_laplace_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.6967346701436833"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_laplace_cdf_complement","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the laplace distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_laplace_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_laplace_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["0.3032653298563167"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_laplace_chf","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the laplace distribution.","examples":[{"description":"","code":"SELECT dist_laplace_chf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_laplace_chf(0, 1.0, 0.5)","align":"left"}],"rows":[["1.1931471805599454"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_laplace_hazard","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the laplace distribution.","examples":[{"description":"","code":"SELECT dist_laplace_hazard(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_laplace_hazard(0, 1.0, 0.5)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_laplace_kurtosis","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the laplace distribution.","examples":[{"description":"","code":"SELECT dist_laplace_kurtosis(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_laplace_kurtosis(0.0, 1.0)","align":"left"}],"rows":[["6.0"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_laplace_kurtosis_excess","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the laplace distribution.","examples":[{"description":"","code":"SELECT dist_laplace_kurtosis_excess(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_laplace_kurtosis_excess(0.0, 1.0)","align":"left"}],"rows":[["3.0"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_laplace_log_cdf","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the laplace distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_laplace_log_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_laplace_log_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.3613506148087591"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_laplace_log_cdf_complement","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the laplace distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_laplace_log_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_laplace_log_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["-1.1931471805599454"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_laplace_log_pdf","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the laplace distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_laplace_log_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_laplace_log_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-1.1931471805599454"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_mean(DOUBLE,DOUBLE)":{"name":"dist_laplace_mean","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the laplace distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_laplace_mean(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_laplace_mean(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_stddev"]},"dist_laplace_median(DOUBLE,DOUBLE)":{"name":"dist_laplace_median","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the laplace distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_laplace_median(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_laplace_median(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_mode(DOUBLE,DOUBLE)":{"name":"dist_laplace_mode","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the laplace distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_laplace_mode(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_laplace_mode(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_laplace_pdf","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the laplace distribution. Returns the probability densityat point x for a laplace distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_laplace_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_laplace_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.3032653298563167"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_laplace_quantile","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the laplace distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_laplace_quantile(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_laplace_quantile(0, 1.0, 0.95)","align":"left"}],"rows":[["2.302585092994045"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_laplace_quantile_complement","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the laplace distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_laplace_quantile_complement(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_laplace_quantile_complement(0, 1.0, 0.95)","align":"left"}],"rows":[["-2.302585092994045"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_range(DOUBLE,DOUBLE)":{"name":"dist_laplace_range","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE[2]","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the laplace distribution.","examples":[{"description":"","code":"SELECT dist_laplace_range(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_laplace_range(0.0, 1.0)","align":"left"}],"rows":[["(-inf, inf)"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_sample(DOUBLE,DOUBLE)":{"name":"dist_laplace_sample","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the laplace distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_laplace_sample(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_laplace_sample(0.0, 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1.0)","align":"left"}],"rows":[["1.4142135623730951"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean"]},"dist_laplace_support(DOUBLE,DOUBLE)":{"name":"dist_laplace_support","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE[2]","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the laplace distribution.","examples":[{"description":"","code":"SELECT dist_laplace_support(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_laplace_support(0.0, 1.0)","align":"left"}],"rows":[["(-inf, inf)"]]}}],"relatedNames":["dist_laplace_sample","dist_laplace_pdf","dist_laplace_cdf","dist_laplace_quantile","dist_laplace_mean","dist_laplace_stddev"]},"dist_laplace_variance(DOUBLE,DOUBLE)":{"name":"dist_laplace_variance","type":"scalar","categories":["Laplace"],"returnType":"DOUBLE","parameters":[{"name":"location","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the laplace distribution.","examples":[{"description":"","code":"SELECT dist_laplace_variance(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_laplace_variance(0.0, 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Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_logistic_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_logistic_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.6224593312018546"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_quantile"]},"dist_logistic_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_logistic_cdf_complement","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the logistic distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_logistic_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_logistic_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["0.3775406687981454"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_logistic_chf","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the logistic distribution.","examples":[{"description":"","code":"SELECT dist_logistic_chf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_logistic_chf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.9740769841801068"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_logistic_hazard","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the logistic distribution.","examples":[{"description":"","code":"SELECT dist_logistic_hazard(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_logistic_hazard(0, 1.0, 0.5)","align":"left"}],"rows":[["0.6224593312018546"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_logistic_kurtosis","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the logistic distribution.","examples":[{"description":"","code":"SELECT dist_logistic_kurtosis(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_logistic_kurtosis(0.0, 1.0)","align":"left"}],"rows":[["4.2"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_logistic_kurtosis_excess","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the logistic distribution.","examples":[{"description":"","code":"SELECT dist_logistic_kurtosis_excess(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_logistic_kurtosis_excess(0.0, 1.0)","align":"left"}],"rows":[["1.2"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_logistic_log_cdf","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the logistic distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_logistic_log_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_logistic_log_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.4740769841801067"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_logistic_log_cdf_complement","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the logistic distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_logistic_log_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_logistic_log_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.9740769841801067"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_logistic_log_pdf","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the logistic distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_logistic_log_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_logistic_log_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-1.4481539683602134"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_median(DOUBLE,DOUBLE)":{"name":"dist_logistic_median","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the logistic distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_logistic_median(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_logistic_median(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_mode(DOUBLE,DOUBLE)":{"name":"dist_logistic_mode","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the logistic distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_logistic_mode(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_logistic_mode(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_logistic_pdf","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the logistic distribution. Returns the probability densityat point x for a logistic distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_logistic_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_logistic_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.2350037122015945"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_logistic_quantile","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the logistic distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_logistic_quantile(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_logistic_quantile(0, 1.0, 0.95)","align":"left"}],"rows":[["2.9444389791664394"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf"]},"dist_logistic_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_logistic_quantile_complement","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the logistic distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_logistic_quantile_complement(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_logistic_quantile_complement(0, 1.0, 0.95)","align":"left"}],"rows":[["-2.9444389791664394"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_range(DOUBLE,DOUBLE)":{"name":"dist_logistic_range","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE[2]","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the logistic distribution.","examples":[{"description":"","code":"SELECT dist_logistic_range(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_logistic_range(0.0, 1.0)","align":"left"}],"rows":[["(-inf, inf)"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_sample(DOUBLE,DOUBLE)":{"name":"dist_logistic_sample","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the logistic distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_logistic_sample(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_logistic_sample(0.0, 1.0)","align":"left"}],"rows":[["-0.17993445860265622"]]}}],"relatedNames":["dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_skewness(DOUBLE,DOUBLE)":{"name":"dist_logistic_skewness","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the logistic distribution.","examples":[{"description":"","code":"SELECT dist_logistic_skewness(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_logistic_skewness(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_support(DOUBLE,DOUBLE)":{"name":"dist_logistic_support","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE[2]","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the logistic distribution.","examples":[{"description":"","code":"SELECT dist_logistic_support(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_logistic_support(0.0, 1.0)","align":"left"}],"rows":[["(-1.7976931348623157e+308, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_logistic_variance(DOUBLE,DOUBLE)":{"name":"dist_logistic_variance","type":"scalar","categories":["Logistic"],"returnType":"DOUBLE","parameters":[{"name":"loc","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the logistic distribution.","examples":[{"description":"","code":"SELECT dist_logistic_variance(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_logistic_variance(0.0, 1.0)","align":"left"}],"rows":[["3.289868133696453"]]}}],"relatedNames":["dist_logistic_sample","dist_logistic_pdf","dist_logistic_cdf","dist_logistic_quantile"]},"dist_lognormal_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_lognormal_cdf","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the lognormal distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_lognormal_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_lognormal_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.24410859578558275"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_lognormal_cdf_complement","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the lognormal distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_lognormal_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_lognormal_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["0.7558914042144173"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_lognormal_chf","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the lognormal distribution.","examples":[{"description":"","code":"SELECT dist_lognormal_chf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_lognormal_chf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.2798575583395914"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_lognormal_hazard","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the lognormal distribution.","examples":[{"description":"","code":"SELECT dist_lognormal_hazard(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_lognormal_hazard(0, 1.0, 0.5)","align":"left"}],"rows":[["0.8301405117419856"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_lognormal_kurtosis","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the lognormal distribution.","examples":[{"description":"","code":"SELECT dist_lognormal_kurtosis(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_kurtosis(0.0, 1.0)","align":"left"}],"rows":[["113.93639217631153"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_lognormal_kurtosis_excess","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the lognormal distribution.","examples":[{"description":"","code":"SELECT dist_lognormal_kurtosis_excess(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_kurtosis_excess(0.0, 1.0)","align":"left"}],"rows":[["110.93639217631153"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_lognormal_log_cdf","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the lognormal distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_lognormal_log_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_lognormal_log_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-1.4101420880058386"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_lognormal_log_cdf_complement","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the lognormal distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_lognormal_log_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_lognormal_log_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.2798575583395914"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_lognormal_log_pdf","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the lognormal distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_lognormal_log_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_lognormal_log_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.46601785960382813"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_mean(DOUBLE,DOUBLE)":{"name":"dist_lognormal_mean","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the lognormal distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_lognormal_mean(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_mean(0.0, 1.0)","align":"left"}],"rows":[["1.6487212707001282"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_stddev"]},"dist_lognormal_median(DOUBLE,DOUBLE)":{"name":"dist_lognormal_median","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the lognormal distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_lognormal_median(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_median(0.0, 1.0)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_mode(DOUBLE,DOUBLE)":{"name":"dist_lognormal_mode","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the lognormal distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_lognormal_mode(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_mode(0.0, 1.0)","align":"left"}],"rows":[["0.36787944117144233"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_lognormal_pdf","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the lognormal distribution. Returns the probability densityat point x for a lognormal distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_lognormal_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_lognormal_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.6274960771159244"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_lognormal_quantile","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the lognormal distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_lognormal_quantile(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_lognormal_quantile(0, 1.0, 0.95)","align":"left"}],"rows":[["5.1802516022330165"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_lognormal_quantile_complement","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the lognormal distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_lognormal_quantile_complement(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_lognormal_quantile_complement(0, 1.0, 0.95)","align":"left"}],"rows":[["0.19304081669873652"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_range(DOUBLE,DOUBLE)":{"name":"dist_lognormal_range","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE[2]","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the lognormal distribution.","examples":[{"description":"","code":"SELECT dist_lognormal_range(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_range(0.0, 1.0)","align":"left"}],"rows":[["(0.0, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_sample(DOUBLE,DOUBLE)":{"name":"dist_lognormal_sample","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the lognormal distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_lognormal_sample(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_sample(0.0, 1.0)","align":"left"}],"rows":[["0.3878694886996256"]]}}],"relatedNames":["dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_skewness(DOUBLE,DOUBLE)":{"name":"dist_lognormal_skewness","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the lognormal distribution.","examples":[{"description":"","code":"SELECT dist_lognormal_skewness(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_skewness(0.0, 1.0)","align":"left"}],"rows":[["6.184877138632554"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_stddev(DOUBLE,DOUBLE)":{"name":"dist_lognormal_stddev","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the standard deviation (σ) of the lognormal distribution.","examples":[{"description":"","code":"SELECT dist_lognormal_stddev(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_stddev(0.0, 1.0)","align":"left"}],"rows":[["2.1611974158950877"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean"]},"dist_lognormal_support(DOUBLE,DOUBLE)":{"name":"dist_lognormal_support","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE[2]","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the lognormal distribution.","examples":[{"description":"","code":"SELECT dist_lognormal_support(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_support(0.0, 1.0)","align":"left"}],"rows":[["(0.0, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_lognormal_variance(DOUBLE,DOUBLE)":{"name":"dist_lognormal_variance","type":"scalar","categories":["Log-normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the lognormal distribution.","examples":[{"description":"","code":"SELECT dist_lognormal_variance(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_lognormal_variance(0.0, 1.0)","align":"left"}],"rows":[["4.670774270471605"]]}}],"relatedNames":["dist_lognormal_sample","dist_lognormal_pdf","dist_lognormal_cdf","dist_lognormal_quantile","dist_lognormal_mean","dist_lognormal_stddev"]},"dist_negative_binomial_cdf(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_negative_binomial_cdf","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the negative_binomial distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_negative_binomial_cdf(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_negative_binomial_cdf(10, 0.5, 5)","align":"left"}],"rows":[["0.15087890624999997"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_cdf_complement(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_negative_binomial_cdf_complement","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the negative_binomial distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_negative_binomial_cdf_complement(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_negative_binomial_cdf_complement(10, 0.5, 5)","align":"left"}],"rows":[["0.84912109375"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_chf(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_negative_binomial_chf","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the negative_binomial distribution.","examples":[{"description":"","code":"SELECT dist_negative_binomial_chf(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_negative_binomial_chf(10, 0.5, 5)","align":"left"}],"rows":[["0.16355347180511537"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_hazard(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_negative_binomial_hazard","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the negative_binomial distribution.","examples":[{"description":"","code":"SELECT dist_negative_binomial_hazard(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_negative_binomial_hazard(10, 0.5, 5)","align":"left"}],"rows":[["0.07195227142035647"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_kurtosis(BIGINT,DOUBLE)":{"name":"dist_negative_binomial_kurtosis","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the negative_binomial distribution.","examples":[{"description":"","code":"SELECT dist_negative_binomial_kurtosis(10, 0.5);","outputTable":{"columns":[{"name":"dist_negative_binomial_kurtosis(10, 0.5)","align":"left"}],"rows":[["3.65"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_kurtosis_excess(BIGINT,DOUBLE)":{"name":"dist_negative_binomial_kurtosis_excess","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the negative_binomial distribution.","examples":[{"description":"","code":"SELECT dist_negative_binomial_kurtosis_excess(10, 0.5);","outputTable":{"columns":[{"name":"dist_negative_binomial_kurtosis_excess(10, 0.5)","align":"left"}],"rows":[["0.65"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_log_cdf(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_negative_binomial_log_cdf","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the negative_binomial distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_negative_binomial_log_cdf(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_negative_binomial_log_cdf(10, 0.5, 5)","align":"left"}],"rows":[["-1.891277709261653"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_log_cdf_complement(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_negative_binomial_log_cdf_complement","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the negative_binomial distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_negative_binomial_log_cdf_complement(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_negative_binomial_log_cdf_complement(10, 0.5, 5)","align":"left"}],"rows":[["-0.16355347180511537"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_log_pdf(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_negative_binomial_log_pdf","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the negative_binomial distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_negative_binomial_log_pdf(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_negative_binomial_log_pdf(10, 0.5, 5)","align":"left"}],"rows":[["-2.7953057485240147"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_median(BIGINT,DOUBLE)":{"name":"dist_negative_binomial_median","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the negative_binomial distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_negative_binomial_median(10, 0.5);","outputTable":{"columns":[{"name":"dist_negative_binomial_median(10, 0.5)","align":"left"}],"rows":[["9.0"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_mode(BIGINT,DOUBLE)":{"name":"dist_negative_binomial_mode","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the negative_binomial distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_negative_binomial_mode(10, 0.5);","outputTable":{"columns":[{"name":"dist_negative_binomial_mode(10, 0.5)","align":"left"}],"rows":[["9.0"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_pdf(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_negative_binomial_pdf","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the negative_binomial distribution. Returns the probability densityat point x for a negative_binomial distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_negative_binomial_pdf(10, 0.5, 5);","outputTable":{"columns":[{"name":"dist_negative_binomial_pdf(10, 0.5, 5)","align":"left"}],"rows":[["0.06109619140624995"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_quantile(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_negative_binomial_quantile","type":"scalar","categories":["Negative Binomial"],"returnType":"BIGINT","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the negative_binomial distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_negative_binomial_quantile(10, 0.5, 0.95);","outputTable":{"columns":[{"name":"dist_negative_binomial_quantile(10, 0.5, 0.95)","align":"left"}],"rows":[["18"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf"]},"dist_negative_binomial_quantile_complement(BIGINT,DOUBLE,DOUBLE)":{"name":"dist_negative_binomial_quantile_complement","type":"scalar","categories":["Negative Binomial"],"returnType":"BIGINT","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the negative_binomial distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_negative_binomial_quantile_complement(10, 0.5, 0.05);","outputTable":{"columns":[{"name":"dist_negative_binomial_quantile_complement(10, 0.5, 0.05)","align":"left"}],"rows":[["18"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_range(BIGINT,DOUBLE)":{"name":"dist_negative_binomial_range","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE[2]","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the negative_binomial distribution.","examples":[{"description":"","code":"SELECT dist_negative_binomial_range(10, 0.5);","outputTable":{"columns":[{"name":"dist_negative_binomial_range(10, 0.5)","align":"left"}],"rows":[["(0.0, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_sample(BIGINT,DOUBLE)":{"name":"dist_negative_binomial_sample","type":"scalar","categories":["Negative Binomial"],"returnType":"BIGINT","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the negative_binomial distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_negative_binomial_sample(10, 0.5);","outputTable":{"columns":[{"name":"dist_negative_binomial_sample(10, 0.5)","align":"left"}],"rows":[["7"]]}}],"relatedNames":["dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_skewness(BIGINT,DOUBLE)":{"name":"dist_negative_binomial_skewness","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the negative_binomial distribution.","examples":[{"description":"","code":"SELECT dist_negative_binomial_skewness(10, 0.5);","outputTable":{"columns":[{"name":"dist_negative_binomial_skewness(10, 0.5)","align":"left"}],"rows":[["0.6708203932499369"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_support(BIGINT,DOUBLE)":{"name":"dist_negative_binomial_support","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE[2]","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the negative_binomial distribution.","examples":[{"description":"","code":"SELECT dist_negative_binomial_support(10, 0.5);","outputTable":{"columns":[{"name":"dist_negative_binomial_support(10, 0.5)","align":"left"}],"rows":[["(0.0, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_negative_binomial_variance(BIGINT,DOUBLE)":{"name":"dist_negative_binomial_variance","type":"scalar","categories":["Negative Binomial"],"returnType":"DOUBLE","parameters":[{"name":"successes","type":"BIGINT","paramType":"positional","description":""},{"name":"prob","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the negative_binomial distribution.","examples":[{"description":"","code":"SELECT dist_negative_binomial_variance(10, 0.5);","outputTable":{"columns":[{"name":"dist_negative_binomial_variance(10, 0.5)","align":"left"}],"rows":[["20.0"]]}}],"relatedNames":["dist_negative_binomial_sample","dist_negative_binomial_pdf","dist_negative_binomial_cdf","dist_negative_binomial_quantile"]},"dist_normal_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_normal_cdf","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the normal distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_normal_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_normal_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.6914624612740131"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_normal_cdf_complement","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the normal distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_normal_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_normal_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["0.3085375387259869"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_normal_chf","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the normal distribution.","examples":[{"description":"","code":"SELECT dist_normal_chf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_normal_chf(0, 1.0, 0.5)","align":"left"}],"rows":[["1.1759117615936188"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_normal_hazard","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the normal distribution.","examples":[{"description":"","code":"SELECT dist_normal_hazard(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_normal_hazard(0, 1.0, 0.5)","align":"left"}],"rows":[["1.1410777703680648"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_normal_kurtosis","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the normal distribution.","examples":[{"description":"","code":"SELECT dist_normal_kurtosis(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_normal_kurtosis(0.0, 1.0)","align":"left"}],"rows":[["3.0"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_normal_kurtosis_excess","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the normal distribution.","examples":[{"description":"","code":"SELECT dist_normal_kurtosis_excess(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_normal_kurtosis_excess(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_normal_log_cdf","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the normal distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_normal_log_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_normal_log_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.36894641528865635"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_normal_log_cdf_complement","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the normal distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_normal_log_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_normal_log_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["-1.1759117615936188"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_normal_log_pdf","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the normal distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_normal_log_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_normal_log_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-1.0439385332046727"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_mean(DOUBLE,DOUBLE)":{"name":"dist_normal_mean","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the normal distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_normal_mean(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_normal_mean(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_stddev"]},"dist_normal_median(DOUBLE,DOUBLE)":{"name":"dist_normal_median","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the normal distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_normal_median(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_normal_median(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_mode(DOUBLE,DOUBLE)":{"name":"dist_normal_mode","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the normal distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_normal_mode(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_normal_mode(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_normal_pdf","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the normal distribution. Returns the probability densityat point x for a normal distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_normal_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_normal_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.3520653267642995"]]}}],"relatedNames":["dist_normal_sample","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_normal_quantile","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the normal distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_normal_quantile(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_normal_quantile(0, 1.0, 0.95)","align":"left"}],"rows":[["1.6448536269514729"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_mean","dist_normal_stddev"]},"dist_normal_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_normal_quantile_complement","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the normal distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_normal_quantile_complement(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_normal_quantile_complement(0, 1.0, 0.95)","align":"left"}],"rows":[["-1.6448536269514729"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_range(DOUBLE,DOUBLE)":{"name":"dist_normal_range","type":"scalar","categories":["Normal"],"returnType":"DOUBLE[2]","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the normal distribution.","examples":[{"description":"","code":"SELECT dist_normal_range(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_normal_range(0.0, 1.0)","align":"left"}],"rows":[["(-inf, inf)"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_sample(DOUBLE,DOUBLE)":{"name":"dist_normal_sample","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the normal distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_normal_sample(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_normal_sample(0.0, 1.0)","align":"left"}],"rows":[["1.5755583945911054"]]}}],"relatedNames":["dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_skewness(DOUBLE,DOUBLE)":{"name":"dist_normal_skewness","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the normal distribution.","examples":[{"description":"","code":"SELECT dist_normal_skewness(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_normal_skewness(0.0, 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1.0)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean"]},"dist_normal_support(DOUBLE,DOUBLE)":{"name":"dist_normal_support","type":"scalar","categories":["Normal"],"returnType":"DOUBLE[2]","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the normal distribution.","examples":[{"description":"","code":"SELECT dist_normal_support(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_normal_support(0.0, 1.0)","align":"left"}],"rows":[["(-inf, inf)"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_normal_variance(DOUBLE,DOUBLE)":{"name":"dist_normal_variance","type":"scalar","categories":["Normal"],"returnType":"DOUBLE","parameters":[{"name":"mean","type":"DOUBLE","paramType":"positional","description":""},{"name":"stddev","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the normal distribution.","examples":[{"description":"","code":"SELECT dist_normal_variance(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_normal_variance(0.0, 1.0)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_normal_sample","dist_normal_pdf","dist_normal_cdf","dist_normal_quantile","dist_normal_mean","dist_normal_stddev"]},"dist_pareto_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_pareto_cdf","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the pareto distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_pareto_cdf(3.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_pareto_cdf(3.0, 1.0, 1.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_pareto_cdf_complement","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the pareto distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_pareto_cdf_complement(3.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_pareto_cdf_complement(3.0, 1.0, 1.5)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_pareto_chf","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the pareto distribution.","examples":[{"description":"","code":"SELECT dist_pareto_chf(3.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_pareto_chf(3.0, 1.0, 1.5)","align":"left"}],"rows":[["-0.0"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_pareto_hazard","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the pareto distribution.","examples":[{"description":"","code":"SELECT dist_pareto_hazard(3.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_pareto_hazard(3.0, 1.0, 1.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_pareto_kurtosis","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the pareto distribution.","examples":[{"description":"","code":"SELECT dist_pareto_kurtosis(3.0, 1.0);"}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_pareto_kurtosis_excess","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the pareto distribution.","examples":[{"description":"","code":"SELECT dist_pareto_kurtosis_excess(3.0, 1.0);"}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_pareto_log_cdf","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the pareto distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_pareto_log_cdf(3.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_pareto_log_cdf(3.0, 1.0, 1.5)","align":"left"}],"rows":[["-inf"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_pareto_log_cdf_complement","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the pareto distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_pareto_log_cdf_complement(3.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_pareto_log_cdf_complement(3.0, 1.0, 1.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_pareto_log_pdf","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the pareto distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_pareto_log_pdf(3.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_pareto_log_pdf(3.0, 1.0, 1.5)","align":"left"}],"rows":[["-inf"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_mean(DOUBLE,DOUBLE)":{"name":"dist_pareto_mean","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the pareto distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_pareto_mean(3.0, 1.0);","outputTable":{"columns":[{"name":"dist_pareto_mean(3.0, 1.0)","align":"left"}],"rows":[["1.7976931348623157e+308"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_stddev"]},"dist_pareto_median(DOUBLE,DOUBLE)":{"name":"dist_pareto_median","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the pareto distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_pareto_median(3.0, 1.0);","outputTable":{"columns":[{"name":"dist_pareto_median(3.0, 1.0)","align":"left"}],"rows":[["6.0"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_mode(DOUBLE,DOUBLE)":{"name":"dist_pareto_mode","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the pareto distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_pareto_mode(3.0, 1.0);","outputTable":{"columns":[{"name":"dist_pareto_mode(3.0, 1.0)","align":"left"}],"rows":[["3.0"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_pareto_pdf","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the pareto distribution. Returns the probability densityat point x for a pareto distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_pareto_pdf(3.0, 1.0, 1.5);","outputTable":{"columns":[{"name":"dist_pareto_pdf(3.0, 1.0, 1.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_pareto_quantile","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the pareto distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_pareto_quantile(3.0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_pareto_quantile(3.0, 1.0, 0.95)","align":"left"}],"rows":[["59.99999999999995"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_pareto_quantile_complement","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the pareto distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_pareto_quantile_complement(3.0, 1.0, 0.05);","outputTable":{"columns":[{"name":"dist_pareto_quantile_complement(3.0, 1.0, 0.05)","align":"left"}],"rows":[["60.0"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_range(DOUBLE,DOUBLE)":{"name":"dist_pareto_range","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE[2]","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the pareto distribution.","examples":[{"description":"","code":"SELECT dist_pareto_range(3.0, 1.0);","outputTable":{"columns":[{"name":"dist_pareto_range(3.0, 1.0)","align":"left"}],"rows":[["(0.0, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_sample(DOUBLE,DOUBLE)":{"name":"dist_pareto_sample","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the pareto distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_pareto_sample(3.0, 1.0);","outputTable":{"columns":[{"name":"dist_pareto_sample(3.0, 1.0)","align":"left"}],"rows":[["3.3830474100303447"]]}}],"relatedNames":["dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_skewness(DOUBLE,DOUBLE)":{"name":"dist_pareto_skewness","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the pareto distribution.","examples":[{"description":"","code":"SELECT dist_pareto_skewness(3.0, 1.0);"}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_stddev(DOUBLE,DOUBLE)":{"name":"dist_pareto_stddev","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the standard deviation (σ) of the pareto distribution.","examples":[{"description":"","code":"SELECT dist_pareto_stddev(3.0, 1.0);"}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean"]},"dist_pareto_support(DOUBLE,DOUBLE)":{"name":"dist_pareto_support","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE[2]","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the pareto distribution.","examples":[{"description":"","code":"SELECT dist_pareto_support(3.0, 1.0);","outputTable":{"columns":[{"name":"dist_pareto_support(3.0, 1.0)","align":"left"}],"rows":[["(3.0, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_pareto_variance(DOUBLE,DOUBLE)":{"name":"dist_pareto_variance","type":"scalar","categories":["Pareto"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"minimum","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the pareto distribution.","examples":[{"description":"","code":"SELECT dist_pareto_variance(3.0, 1.0);"}],"relatedNames":["dist_pareto_sample","dist_pareto_pdf","dist_pareto_cdf","dist_pareto_quantile","dist_pareto_mean","dist_pareto_stddev"]},"dist_poisson_cdf(DOUBLE,DOUBLE)":{"name":"dist_poisson_cdf","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the poisson distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_poisson_cdf(5.0, 3);","outputTable":{"columns":[{"name":"dist_poisson_cdf(5.0, 3)","align":"left"}],"rows":[["0.2650259152973617"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_cdf_complement(DOUBLE,DOUBLE)":{"name":"dist_poisson_cdf_complement","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the poisson distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_poisson_cdf_complement(5.0, 3);","outputTable":{"columns":[{"name":"dist_poisson_cdf_complement(5.0, 3)","align":"left"}],"rows":[["0.7349740847026383"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_chf(DOUBLE,DOUBLE)":{"name":"dist_poisson_chf","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the poisson distribution.","examples":[{"description":"","code":"SELECT dist_poisson_chf(5.0, 3);","outputTable":{"columns":[{"name":"dist_poisson_chf(5.0, 3)","align":"left"}],"rows":[["0.30792003929888545"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_hazard(DOUBLE,DOUBLE)":{"name":"dist_poisson_hazard","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the poisson distribution.","examples":[{"description":"","code":"SELECT dist_poisson_hazard(5.0, 3);","outputTable":{"columns":[{"name":"dist_poisson_hazard(5.0, 3)","align":"left"}],"rows":[["0.1909916264205072"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_kurtosis(DOUBLE)":{"name":"dist_poisson_kurtosis","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the poisson distribution.","examples":[{"description":"","code":"SELECT dist_poisson_kurtosis(5.0);","outputTable":{"columns":[{"name":"dist_poisson_kurtosis(5.0)","align":"left"}],"rows":[["3.2"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_kurtosis_excess(DOUBLE)":{"name":"dist_poisson_kurtosis_excess","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the poisson distribution.","examples":[{"description":"","code":"SELECT dist_poisson_kurtosis_excess(5.0);","outputTable":{"columns":[{"name":"dist_poisson_kurtosis_excess(5.0)","align":"left"}],"rows":[["0.2"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_log_cdf(DOUBLE,DOUBLE)":{"name":"dist_poisson_log_cdf","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the poisson distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_poisson_log_cdf(5.0, 3);","outputTable":{"columns":[{"name":"dist_poisson_log_cdf(5.0, 3)","align":"left"}],"rows":[["-1.327927664202445"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_log_cdf_complement(DOUBLE,DOUBLE)":{"name":"dist_poisson_log_cdf_complement","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the poisson distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_poisson_log_cdf_complement(5.0, 3);","outputTable":{"columns":[{"name":"dist_poisson_log_cdf_complement(5.0, 3)","align":"left"}],"rows":[["-0.30792003929888545"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_log_pdf(DOUBLE,DOUBLE)":{"name":"dist_poisson_log_pdf","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the poisson distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_poisson_log_pdf(5.0, 3);","outputTable":{"columns":[{"name":"dist_poisson_log_pdf(5.0, 3)","align":"left"}],"rows":[["-1.963445731925754"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_mean(DOUBLE)":{"name":"dist_poisson_mean","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the poisson distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_poisson_mean(5.0);","outputTable":{"columns":[{"name":"dist_poisson_mean(5.0)","align":"left"}],"rows":[["5.0"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_stddev"]},"dist_poisson_median(DOUBLE)":{"name":"dist_poisson_median","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the poisson distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_poisson_median(5.0);","outputTable":{"columns":[{"name":"dist_poisson_median(5.0)","align":"left"}],"rows":[["5.0"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_mode(DOUBLE)":{"name":"dist_poisson_mode","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the poisson distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_poisson_mode(5.0);","outputTable":{"columns":[{"name":"dist_poisson_mode(5.0)","align":"left"}],"rows":[["5.0"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_pdf(DOUBLE,DOUBLE)":{"name":"dist_poisson_pdf","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the poisson distribution. Returns the probability densityat point x for a poisson distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_poisson_pdf(5.0, 3);","outputTable":{"columns":[{"name":"dist_poisson_pdf(5.0, 3)","align":"left"}],"rows":[["0.1403738958142805"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_quantile(DOUBLE,DOUBLE)":{"name":"dist_poisson_quantile","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the poisson distribution. 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Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_poisson_quantile_complement(5.0, 0.05);","outputTable":{"columns":[{"name":"dist_poisson_quantile_complement(5.0, 0.05)","align":"left"}],"rows":[["9.0"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_range(DOUBLE)":{"name":"dist_poisson_range","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE[2]","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the poisson distribution.","examples":[{"description":"","code":"SELECT dist_poisson_range(5.0);","outputTable":{"columns":[{"name":"dist_poisson_range(5.0)","align":"left"}],"rows":[["(0.0, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_sample(DOUBLE)":{"name":"dist_poisson_sample","type":"scalar","categories":["Poisson"],"returnType":"BIGINT","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the poisson distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_poisson_sample(5.0);","outputTable":{"columns":[{"name":"dist_poisson_sample(5.0)","align":"left"}],"rows":[["5"]]}}],"relatedNames":["dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_skewness(DOUBLE)":{"name":"dist_poisson_skewness","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the poisson distribution.","examples":[{"description":"","code":"SELECT dist_poisson_skewness(5.0);","outputTable":{"columns":[{"name":"dist_poisson_skewness(5.0)","align":"left"}],"rows":[["0.4472135954999579"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_stddev(DOUBLE)":{"name":"dist_poisson_stddev","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the standard deviation (σ) of the poisson distribution.","examples":[{"description":"","code":"SELECT dist_poisson_stddev(5.0);","outputTable":{"columns":[{"name":"dist_poisson_stddev(5.0)","align":"left"}],"rows":[["2.23606797749979"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean"]},"dist_poisson_support(DOUBLE)":{"name":"dist_poisson_support","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE[2]","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the poisson distribution.","examples":[{"description":"","code":"SELECT dist_poisson_support(5.0);","outputTable":{"columns":[{"name":"dist_poisson_support(5.0)","align":"left"}],"rows":[["(0.0, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_poisson_variance(DOUBLE)":{"name":"dist_poisson_variance","type":"scalar","categories":["Poisson"],"returnType":"DOUBLE","parameters":[{"name":"rate","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the poisson distribution.","examples":[{"description":"","code":"SELECT dist_poisson_variance(5.0);","outputTable":{"columns":[{"name":"dist_poisson_variance(5.0)","align":"left"}],"rows":[["5.0"]]}}],"relatedNames":["dist_poisson_sample","dist_poisson_pdf","dist_poisson_cdf","dist_poisson_quantile","dist_poisson_mean","dist_poisson_stddev"]},"dist_rayleigh_cdf(DOUBLE,DOUBLE)":{"name":"dist_rayleigh_cdf","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the rayleigh distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_rayleigh_cdf(1.0, 0.5);","outputTable":{"columns":[{"name":"dist_rayleigh_cdf(1.0, 0.5)","align":"left"}],"rows":[["0.1175030974154046"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_cdf_complement(DOUBLE,DOUBLE)":{"name":"dist_rayleigh_cdf_complement","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the rayleigh distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_rayleigh_cdf_complement(1.0, 0.5);","outputTable":{"columns":[{"name":"dist_rayleigh_cdf_complement(1.0, 0.5)","align":"left"}],"rows":[["0.8824969025845955"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_chf(DOUBLE,DOUBLE)":{"name":"dist_rayleigh_chf","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the rayleigh distribution.","examples":[{"description":"","code":"SELECT dist_rayleigh_chf(1.0, 0.5);","outputTable":{"columns":[{"name":"dist_rayleigh_chf(1.0, 0.5)","align":"left"}],"rows":[["0.12499999999999994"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_hazard(DOUBLE,DOUBLE)":{"name":"dist_rayleigh_hazard","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the rayleigh distribution.","examples":[{"description":"","code":"SELECT dist_rayleigh_hazard(1.0, 0.5);","outputTable":{"columns":[{"name":"dist_rayleigh_hazard(1.0, 0.5)","align":"left"}],"rows":[["0.5"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_kurtosis(DOUBLE)":{"name":"dist_rayleigh_kurtosis","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the rayleigh distribution.","examples":[{"description":"","code":"SELECT dist_rayleigh_kurtosis(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_kurtosis(1.0)","align":"left"}],"rows":[["3.245089300687638"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_kurtosis_excess(DOUBLE)":{"name":"dist_rayleigh_kurtosis_excess","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the rayleigh distribution.","examples":[{"description":"","code":"SELECT dist_rayleigh_kurtosis_excess(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_kurtosis_excess(1.0)","align":"left"}],"rows":[["0.24508930068763807"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_log_cdf(DOUBLE,DOUBLE)":{"name":"dist_rayleigh_log_cdf","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the rayleigh distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_rayleigh_log_cdf(1.0, 0.5);","outputTable":{"columns":[{"name":"dist_rayleigh_log_cdf(1.0, 0.5)","align":"left"}],"rows":[["-2.1412905847632016"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_log_cdf_complement(DOUBLE,DOUBLE)":{"name":"dist_rayleigh_log_cdf_complement","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the rayleigh distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_rayleigh_log_cdf_complement(1.0, 0.5);","outputTable":{"columns":[{"name":"dist_rayleigh_log_cdf_complement(1.0, 0.5)","align":"left"}],"rows":[["-0.125"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_log_pdf(DOUBLE,DOUBLE)":{"name":"dist_rayleigh_log_pdf","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the rayleigh distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_rayleigh_log_pdf(1.0, 0.5);","outputTable":{"columns":[{"name":"dist_rayleigh_log_pdf(1.0, 0.5)","align":"left"}],"rows":[["-0.8181471805599453"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_mean(DOUBLE)":{"name":"dist_rayleigh_mean","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the rayleigh distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_rayleigh_mean(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_mean(1.0)","align":"left"}],"rows":[["1.2533141373155003"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_stddev"]},"dist_rayleigh_median(DOUBLE)":{"name":"dist_rayleigh_median","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the rayleigh distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_rayleigh_median(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_median(1.0)","align":"left"}],"rows":[["1.1774100225154747"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_mode(DOUBLE)":{"name":"dist_rayleigh_mode","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the rayleigh distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_rayleigh_mode(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_mode(1.0)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_pdf(DOUBLE,DOUBLE)":{"name":"dist_rayleigh_pdf","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the rayleigh distribution. Returns the probability densityat point x for a rayleigh distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_rayleigh_pdf(1.0, 0.5);","outputTable":{"columns":[{"name":"dist_rayleigh_pdf(1.0, 0.5)","align":"left"}],"rows":[["0.4412484512922977"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_quantile(DOUBLE,DOUBLE)":{"name":"dist_rayleigh_quantile","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the rayleigh distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_rayleigh_quantile(1.0, 0.95);","outputTable":{"columns":[{"name":"dist_rayleigh_quantile(1.0, 0.95)","align":"left"}],"rows":[["2.447746830680816"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_quantile_complement(DOUBLE,DOUBLE)":{"name":"dist_rayleigh_quantile_complement","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the rayleigh distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_rayleigh_quantile_complement(1.0, 0.05);","outputTable":{"columns":[{"name":"dist_rayleigh_quantile_complement(1.0, 0.05)","align":"left"}],"rows":[["2.4477468306808166"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_range(DOUBLE)":{"name":"dist_rayleigh_range","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE[2]","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the rayleigh distribution.","examples":[{"description":"","code":"SELECT dist_rayleigh_range(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_range(1.0)","align":"left"}],"rows":[["(0.0, inf)"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_sample(DOUBLE)":{"name":"dist_rayleigh_sample","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the rayleigh distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_rayleigh_sample(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_sample(1.0)","align":"left"}],"rows":[["1.483969980665358"]]}}],"relatedNames":["dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_skewness(DOUBLE)":{"name":"dist_rayleigh_skewness","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the rayleigh distribution.","examples":[{"description":"","code":"SELECT dist_rayleigh_skewness(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_skewness(1.0)","align":"left"}],"rows":[["0.6311106578189372"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_stddev(DOUBLE)":{"name":"dist_rayleigh_stddev","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the standard deviation (σ) of the rayleigh distribution.","examples":[{"description":"","code":"SELECT dist_rayleigh_stddev(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_stddev(1.0)","align":"left"}],"rows":[["0.6551363775620336"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean"]},"dist_rayleigh_support(DOUBLE)":{"name":"dist_rayleigh_support","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE[2]","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the rayleigh distribution.","examples":[{"description":"","code":"SELECT dist_rayleigh_support(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_support(1.0)","align":"left"}],"rows":[["(0.0, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_rayleigh_variance(DOUBLE)":{"name":"dist_rayleigh_variance","type":"scalar","categories":["Rayleigh"],"returnType":"DOUBLE","parameters":[{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the rayleigh distribution.","examples":[{"description":"","code":"SELECT dist_rayleigh_variance(1.0);","outputTable":{"columns":[{"name":"dist_rayleigh_variance(1.0)","align":"left"}],"rows":[["0.4292036732051034"]]}}],"relatedNames":["dist_rayleigh_sample","dist_rayleigh_pdf","dist_rayleigh_cdf","dist_rayleigh_quantile","dist_rayleigh_mean","dist_rayleigh_stddev"]},"dist_students_t_cdf(DOUBLE,DOUBLE)":{"name":"dist_students_t_cdf","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the students_t distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_students_t_cdf(10, 1.5);","outputTable":{"columns":[{"name":"dist_students_t_cdf(10, 1.5)","align":"left"}],"rows":[["0.9177463367772799"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_cdf_complement(DOUBLE,DOUBLE)":{"name":"dist_students_t_cdf_complement","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the students_t distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_students_t_cdf_complement(10, 1.5);","outputTable":{"columns":[{"name":"dist_students_t_cdf_complement(10, 1.5)","align":"left"}],"rows":[["0.08225366322272007"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_chf(DOUBLE,DOUBLE)":{"name":"dist_students_t_chf","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the students_t distribution.","examples":[{"description":"","code":"SELECT dist_students_t_chf(10, 1.5);","outputTable":{"columns":[{"name":"dist_students_t_chf(10, 1.5)","align":"left"}],"rows":[["2.497947352666178"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_hazard(DOUBLE,DOUBLE)":{"name":"dist_students_t_hazard","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the students_t distribution.","examples":[{"description":"","code":"SELECT dist_students_t_hazard(10, 1.5);","outputTable":{"columns":[{"name":"dist_students_t_hazard(10, 1.5)","align":"left"}],"rows":[["1.5494117744276812"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_kurtosis(DOUBLE)":{"name":"dist_students_t_kurtosis","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the students_t distribution.","examples":[{"description":"","code":"SELECT dist_students_t_kurtosis(10);","outputTable":{"columns":[{"name":"dist_students_t_kurtosis(10)","align":"left"}],"rows":[["4.0"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_kurtosis_excess(DOUBLE)":{"name":"dist_students_t_kurtosis_excess","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the students_t distribution.","examples":[{"description":"","code":"SELECT dist_students_t_kurtosis_excess(10);","outputTable":{"columns":[{"name":"dist_students_t_kurtosis_excess(10)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_log_cdf(DOUBLE,DOUBLE)":{"name":"dist_students_t_log_cdf","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the students_t distribution. 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Returns the probability densityat point x for a students_t distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_students_t_pdf(10, 1.5);","outputTable":{"columns":[{"name":"dist_students_t_pdf(10, 1.5)","align":"left"}],"rows":[["0.1274447942870916"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_quantile(DOUBLE,DOUBLE)":{"name":"dist_students_t_quantile","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the students_t distribution. 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Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_students_t_quantile_complement(10, 0.05);","outputTable":{"columns":[{"name":"dist_students_t_quantile_complement(10, 0.05)","align":"left"}],"rows":[["1.8124611228116767"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_range(DOUBLE)":{"name":"dist_students_t_range","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE[2]","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the students_t distribution.","examples":[{"description":"","code":"SELECT dist_students_t_range(10);","outputTable":{"columns":[{"name":"dist_students_t_range(10)","align":"left"}],"rows":[["(-inf, inf)"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_sample(DOUBLE)":{"name":"dist_students_t_sample","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the students_t distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_students_t_sample(10);","outputTable":{"columns":[{"name":"dist_students_t_sample(10)","align":"left"}],"rows":[["-0.5258729282048364"]]}}],"relatedNames":["dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_skewness(DOUBLE)":{"name":"dist_students_t_skewness","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the students_t distribution.","examples":[{"description":"","code":"SELECT dist_students_t_skewness(10);","outputTable":{"columns":[{"name":"dist_students_t_skewness(10)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_stddev(DOUBLE)":{"name":"dist_students_t_stddev","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the standard deviation (σ) of the students_t distribution.","examples":[{"description":"","code":"SELECT dist_students_t_stddev(10);","outputTable":{"columns":[{"name":"dist_students_t_stddev(10)","align":"left"}],"rows":[["1.118033988749895"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean"]},"dist_students_t_support(DOUBLE)":{"name":"dist_students_t_support","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE[2]","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the students_t distribution.","examples":[{"description":"","code":"SELECT dist_students_t_support(10);","outputTable":{"columns":[{"name":"dist_students_t_support(10)","align":"left"}],"rows":[["(-inf, inf)"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_students_t_variance(DOUBLE)":{"name":"dist_students_t_variance","type":"scalar","categories":["Student's t"],"returnType":"DOUBLE","parameters":[{"name":"degrees_of_freedom","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the students_t distribution.","examples":[{"description":"","code":"SELECT dist_students_t_variance(10);","outputTable":{"columns":[{"name":"dist_students_t_variance(10)","align":"left"}],"rows":[["1.25"]]}}],"relatedNames":["dist_students_t_sample","dist_students_t_pdf","dist_students_t_cdf","dist_students_t_quantile","dist_students_t_mean","dist_students_t_stddev"]},"dist_uniform_int_cdf(DOUBLE,DOUBLE,BIGINT)":{"name":"dist_uniform_int_cdf","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"BIGINT","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the uniform_int distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_uniform_int_cdf(1, 6, 3);","outputTable":{"columns":[{"name":"dist_uniform_int_cdf(1, 6, 3)","align":"left"}],"rows":[["0.4"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_cdf_complement(DOUBLE,DOUBLE,BIGINT)":{"name":"dist_uniform_int_cdf_complement","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"BIGINT","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the uniform_int distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_uniform_int_cdf_complement(1, 6, 3);","outputTable":{"columns":[{"name":"dist_uniform_int_cdf_complement(1, 6, 3)","align":"left"}],"rows":[["0.6"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_chf(DOUBLE,DOUBLE,BIGINT)":{"name":"dist_uniform_int_chf","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"BIGINT","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the uniform_int distribution.","examples":[{"description":"","code":"SELECT dist_uniform_int_chf(1, 6, 3);","outputTable":{"columns":[{"name":"dist_uniform_int_chf(1, 6, 3)","align":"left"}],"rows":[["0.5108256237659907"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_hazard(DOUBLE,DOUBLE,BIGINT)":{"name":"dist_uniform_int_hazard","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"BIGINT","paramType":"positional","description":""}],"description":"Computes the hazard function of the uniform_int distribution.","examples":[{"description":"","code":"SELECT dist_uniform_int_hazard(1, 6, 3);","outputTable":{"columns":[{"name":"dist_uniform_int_hazard(1, 6, 3)","align":"left"}],"rows":[["0.33333333333333337"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_uniform_int_kurtosis","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the uniform_int distribution.","examples":[{"description":"","code":"SELECT dist_uniform_int_kurtosis(1, 6);","outputTable":{"columns":[{"name":"dist_uniform_int_kurtosis(1, 6)","align":"left"}],"rows":[["1.8"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_uniform_int_kurtosis_excess","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the uniform_int distribution.","examples":[{"description":"","code":"SELECT dist_uniform_int_kurtosis_excess(1, 6);","outputTable":{"columns":[{"name":"dist_uniform_int_kurtosis_excess(1, 6)","align":"left"}],"rows":[["-1.2"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_log_cdf(DOUBLE,DOUBLE,BIGINT)":{"name":"dist_uniform_int_log_cdf","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"BIGINT","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the uniform_int distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_uniform_int_log_cdf(1, 6, 3);","outputTable":{"columns":[{"name":"dist_uniform_int_log_cdf(1, 6, 3)","align":"left"}],"rows":[["-0.916290731874155"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_log_cdf_complement(DOUBLE,DOUBLE,BIGINT)":{"name":"dist_uniform_int_log_cdf_complement","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"BIGINT","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the uniform_int distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_uniform_int_log_cdf_complement(1, 6, 3);","outputTable":{"columns":[{"name":"dist_uniform_int_log_cdf_complement(1, 6, 3)","align":"left"}],"rows":[["-0.5108256237659907"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_log_pdf(DOUBLE,DOUBLE,BIGINT)":{"name":"dist_uniform_int_log_pdf","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"BIGINT","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the uniform_int distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_uniform_int_log_pdf(1, 6, 3);","outputTable":{"columns":[{"name":"dist_uniform_int_log_pdf(1, 6, 3)","align":"left"}],"rows":[["-1.6094379124341003"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_mean(DOUBLE,DOUBLE)":{"name":"dist_uniform_int_mean","type":"scalar","categories":["Uniform (Integer)"],"returnType":"BIGINT","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the uniform_int distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_uniform_int_mean(1, 6);","outputTable":{"columns":[{"name":"dist_uniform_int_mean(1, 6)","align":"left"}],"rows":[["3"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_stddev"]},"dist_uniform_int_median(DOUBLE,DOUBLE)":{"name":"dist_uniform_int_median","type":"scalar","categories":["Uniform (Integer)"],"returnType":"BIGINT","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the uniform_int distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_uniform_int_median(1, 6);","outputTable":{"columns":[{"name":"dist_uniform_int_median(1, 6)","align":"left"}],"rows":[["3"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_mode(DOUBLE,DOUBLE)":{"name":"dist_uniform_int_mode","type":"scalar","categories":["Uniform (Integer)"],"returnType":"BIGINT","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the uniform_int distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_uniform_int_mode(1, 6);","outputTable":{"columns":[{"name":"dist_uniform_int_mode(1, 6)","align":"left"}],"rows":[["1"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_pdf(DOUBLE,DOUBLE,BIGINT)":{"name":"dist_uniform_int_pdf","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"BIGINT","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the uniform_int distribution. Returns the probability densityat point x for a uniform_int distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_uniform_int_pdf(1, 6, 3);","outputTable":{"columns":[{"name":"dist_uniform_int_pdf(1, 6, 3)","align":"left"}],"rows":[["0.2"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_int_quantile","type":"scalar","categories":["Uniform (Integer)"],"returnType":"BIGINT","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the uniform_int distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_uniform_int_quantile(1, 6, 0.95);","outputTable":{"columns":[{"name":"dist_uniform_int_quantile(1, 6, 0.95)","align":"left"}],"rows":[["5"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_int_quantile_complement","type":"scalar","categories":["Uniform (Integer)"],"returnType":"BIGINT","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the uniform_int distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_uniform_int_quantile_complement(1, 6, 0.05);","outputTable":{"columns":[{"name":"dist_uniform_int_quantile_complement(1, 6, 0.05)","align":"left"}],"rows":[["5"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_range(DOUBLE,DOUBLE)":{"name":"dist_uniform_int_range","type":"scalar","categories":["Uniform (Integer)"],"returnType":"BIGINT[2]","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the uniform_int distribution.","examples":[{"description":"","code":"SELECT dist_uniform_int_range(1, 6);","outputTable":{"columns":[{"name":"dist_uniform_int_range(1, 6)","align":"left"}],"rows":[["(0, 4621819117588971750)"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_sample(DOUBLE,DOUBLE)":{"name":"dist_uniform_int_sample","type":"scalar","categories":["Uniform (Integer)"],"returnType":"BIGINT","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the uniform_int distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_uniform_int_sample(1, 6);","outputTable":{"columns":[{"name":"dist_uniform_int_sample(1, 6)","align":"left"}],"rows":[["1"]]}}],"relatedNames":["dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_skewness(DOUBLE,DOUBLE)":{"name":"dist_uniform_int_skewness","type":"scalar","categories":["Uniform (Integer)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the skewness of the uniform_int distribution.","examples":[{"description":"","code":"SELECT dist_uniform_int_skewness(1, 6);","outputTable":{"columns":[{"name":"dist_uniform_int_skewness(1, 6)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_stddev(DOUBLE,DOUBLE)":{"name":"dist_uniform_int_stddev","type":"scalar","categories":["Uniform (Integer)"],"returnType":"BIGINT","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the standard deviation (σ) of the uniform_int distribution.","examples":[{"description":"","code":"SELECT dist_uniform_int_stddev(1, 6);","outputTable":{"columns":[{"name":"dist_uniform_int_stddev(1, 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6)"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_int_variance(DOUBLE,DOUBLE)":{"name":"dist_uniform_int_variance","type":"scalar","categories":["Uniform (Integer)"],"returnType":"BIGINT","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the uniform_int distribution.","examples":[{"description":"","code":"SELECT dist_uniform_int_variance(1, 6);","outputTable":{"columns":[{"name":"dist_uniform_int_variance(1, 6)","align":"left"}],"rows":[["2"]]}}],"relatedNames":["dist_uniform_int_sample","dist_uniform_int_pdf","dist_uniform_int_cdf","dist_uniform_int_quantile","dist_uniform_int_mean","dist_uniform_int_stddev"]},"dist_uniform_real_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_real_cdf","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the uniform_real distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_uniform_real_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_uniform_real_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.5"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_real_cdf_complement","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the uniform_real distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_uniform_real_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_uniform_real_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["0.5"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_real_chf","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the uniform_real distribution.","examples":[{"description":"","code":"SELECT dist_uniform_real_chf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_uniform_real_chf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.6931471805599453"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_real_hazard","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the uniform_real distribution.","examples":[{"description":"","code":"SELECT dist_uniform_real_hazard(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_uniform_real_hazard(0, 1.0, 0.5)","align":"left"}],"rows":[["2.0"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_uniform_real_kurtosis","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the uniform_real distribution.","examples":[{"description":"","code":"SELECT dist_uniform_real_kurtosis(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_uniform_real_kurtosis(0.0, 1.0)","align":"left"}],"rows":[["1.8"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_uniform_real_kurtosis_excess","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the uniform_real distribution.","examples":[{"description":"","code":"SELECT dist_uniform_real_kurtosis_excess(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_uniform_real_kurtosis_excess(0.0, 1.0)","align":"left"}],"rows":[["-1.2"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_real_log_cdf","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the uniform_real distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_uniform_real_log_cdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_uniform_real_log_cdf(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.6931471805599453"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_real_log_cdf_complement","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the uniform_real distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_uniform_real_log_cdf_complement(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_uniform_real_log_cdf_complement(0, 1.0, 0.5)","align":"left"}],"rows":[["-0.6931471805599453"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_real_log_pdf","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the uniform_real distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_uniform_real_log_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_uniform_real_log_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_mean(DOUBLE,DOUBLE)":{"name":"dist_uniform_real_mean","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mean (μ) of the uniform_real distribution, which is the first moment.","examples":[{"description":"","code":"SELECT dist_uniform_real_mean(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_uniform_real_mean(0.0, 1.0)","align":"left"}],"rows":[["0.5"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_stddev"]},"dist_uniform_real_median(DOUBLE,DOUBLE)":{"name":"dist_uniform_real_median","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the uniform_real distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_uniform_real_median(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_uniform_real_median(0.0, 1.0)","align":"left"}],"rows":[["0.5"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_mode(DOUBLE,DOUBLE)":{"name":"dist_uniform_real_mode","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the uniform_real distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_uniform_real_mode(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_uniform_real_mode(0.0, 1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_real_pdf","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the uniform_real distribution. Returns the probability densityat point x for a uniform_real distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_uniform_real_pdf(0, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_uniform_real_pdf(0, 1.0, 0.5)","align":"left"}],"rows":[["1.0"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_real_quantile","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the uniform_real distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_uniform_real_quantile(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_uniform_real_quantile(0, 1.0, 0.95)","align":"left"}],"rows":[["0.95"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_uniform_real_quantile_complement","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the uniform_real distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_uniform_real_quantile_complement(0, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_uniform_real_quantile_complement(0, 1.0, 0.95)","align":"left"}],"rows":[["0.050000000000000044"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_range(DOUBLE,DOUBLE)":{"name":"dist_uniform_real_range","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE[2]","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the uniform_real distribution.","examples":[{"description":"","code":"SELECT dist_uniform_real_range(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_uniform_real_range(0.0, 1.0)","align":"left"}],"rows":[["(-1.7976931348623157e+308, 1.7976931348623157e+308)"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_sample(DOUBLE,DOUBLE)":{"name":"dist_uniform_real_sample","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Generates random samples from the uniform_real distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_uniform_real_sample(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_uniform_real_sample(0.0, 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1.0)","align":"left"}],"rows":[["0.0"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_stddev(DOUBLE,DOUBLE)":{"name":"dist_uniform_real_stddev","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the standard deviation (σ) of the uniform_real distribution.","examples":[{"description":"","code":"SELECT dist_uniform_real_stddev(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_uniform_real_stddev(0.0, 1.0)","align":"left"}],"rows":[["0.28867513459481287"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean"]},"dist_uniform_real_support(DOUBLE,DOUBLE)":{"name":"dist_uniform_real_support","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE[2]","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the support of the uniform_real distribution.","examples":[{"description":"","code":"SELECT dist_uniform_real_support(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_uniform_real_support(0.0, 1.0)","align":"left"}],"rows":[["(0.0, 1.0)"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_uniform_real_variance(DOUBLE,DOUBLE)":{"name":"dist_uniform_real_variance","type":"scalar","categories":["Uniform (Real)"],"returnType":"DOUBLE","parameters":[{"name":"min","type":"DOUBLE","paramType":"positional","description":""},{"name":"max","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the variance (σ²) of the uniform_real distribution.","examples":[{"description":"","code":"SELECT dist_uniform_real_variance(0.0, 1.0);","outputTable":{"columns":[{"name":"dist_uniform_real_variance(0.0, 1.0)","align":"left"}],"rows":[["0.08333333333333333"]]}}],"relatedNames":["dist_uniform_real_sample","dist_uniform_real_pdf","dist_uniform_real_cdf","dist_uniform_real_quantile","dist_uniform_real_mean","dist_uniform_real_stddev"]},"dist_weibull_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_weibull_cdf","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative distribution function (CDF) of the weibull distribution. Returns the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_weibull_cdf(1.5, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_weibull_cdf(1.5, 1.0, 0.5)","align":"left"}],"rows":[["0.2978114986734404"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_quantile"]},"dist_weibull_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_weibull_cdf_complement","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary cumulative distribution function (1 - CDF) of the weibull distribution. Returns the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_weibull_cdf_complement(1.5, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_weibull_cdf_complement(1.5, 1.0, 0.5)","align":"left"}],"rows":[["0.7021885013265596"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_chf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_weibull_chf","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the cumulative hazard function of the weibull distribution.","examples":[{"description":"","code":"SELECT dist_weibull_chf(1.5, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_weibull_chf(1.5, 1.0, 0.5)","align":"left"}],"rows":[["0.35355339059327373"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_hazard(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_weibull_hazard","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the hazard function of the weibull distribution.","examples":[{"description":"","code":"SELECT dist_weibull_hazard(1.5, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_weibull_hazard(1.5, 1.0, 0.5)","align":"left"}],"rows":[["1.0606601717798214"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_kurtosis(DOUBLE,DOUBLE)":{"name":"dist_weibull_kurtosis","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the kurtosis of the weibull distribution.","examples":[{"description":"","code":"SELECT dist_weibull_kurtosis(1.5, 1.0);","outputTable":{"columns":[{"name":"dist_weibull_kurtosis(1.5, 1.0)","align":"left"}],"rows":[["4.3904035615957975"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_kurtosis_excess(DOUBLE,DOUBLE)":{"name":"dist_weibull_kurtosis_excess","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the excess kurtosis of the weibull distribution.","examples":[{"description":"","code":"SELECT dist_weibull_kurtosis_excess(1.5, 1.0);","outputTable":{"columns":[{"name":"dist_weibull_kurtosis_excess(1.5, 1.0)","align":"left"}],"rows":[["1.3904035615957977"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_log_cdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_weibull_log_cdf","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the cumulative distribution function (CDF) of the weibull distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.","examples":[{"description":"","code":"SELECT dist_weibull_log_cdf(1.5, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_weibull_log_cdf(1.5, 1.0, 0.5)","align":"left"}],"rows":[["-1.211294547411032"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_weibull_log_cdf_complement","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the weibull distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.","examples":[{"description":"","code":"SELECT dist_weibull_log_cdf_complement(1.5, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_weibull_log_cdf_complement(1.5, 1.0, 0.5)","align":"left"}],"rows":[["-0.3535533905932738"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_log_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_weibull_log_pdf","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the natural logarithm of the probability density function (log-PDF) of the weibull distribution. Useful for numerical stability when dealing with very small probabilities.","examples":[{"description":"","code":"SELECT dist_weibull_log_pdf(1.5, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_weibull_log_pdf(1.5, 1.0, 0.5)","align":"left"}],"rows":[["-0.29466187276508204"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_median(DOUBLE,DOUBLE)":{"name":"dist_weibull_median","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the median (50th percentile) of the weibull distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_weibull_median(1.5, 1.0);","outputTable":{"columns":[{"name":"dist_weibull_median(1.5, 1.0)","align":"left"}],"rows":[["0.7832197687746514"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_mode(DOUBLE,DOUBLE)":{"name":"dist_weibull_mode","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the mode (most likely value) of the weibull distribution, which equals the mean.","examples":[{"description":"","code":"SELECT dist_weibull_mode(1.5, 1.0);","outputTable":{"columns":[{"name":"dist_weibull_mode(1.5, 1.0)","align":"left"}],"rows":[["0.4807498567691361"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_pdf(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_weibull_pdf","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"x","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the probability density function (PDF) of the weibull distribution. Returns the probability densityat point x for a weibull distribution with specified parameters.","examples":[{"description":"","code":"SELECT dist_weibull_pdf(1.5, 1.0, 0.5);","outputTable":{"columns":[{"name":"dist_weibull_pdf(1.5, 1.0, 0.5)","align":"left"}],"rows":[["0.7447833764388441"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_quantile(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_weibull_quantile","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the quantile function (inverse CDF) of the weibull distribution. Returns the value x such that P(X ≤ x) = p, where p is the cumulative probability.","examples":[{"description":"","code":"SELECT dist_weibull_quantile(1.5, 1.0, 0.95);","outputTable":{"columns":[{"name":"dist_weibull_quantile(1.5, 1.0, 0.95)","align":"left"}],"rows":[["2.0781106375345564"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf"]},"dist_weibull_quantile_complement(DOUBLE,DOUBLE,DOUBLE)":{"name":"dist_weibull_quantile_complement","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""},{"name":"p","type":"DOUBLE","paramType":"positional","description":""}],"description":"Computes the complementary quantile function of the weibull distribution. Returns the value x such that P(X > x) = p, useful for computing upper tail quantiles.","examples":[{"description":"","code":"SELECT dist_weibull_quantile_complement(1.5, 1.0, 0.05);","outputTable":{"columns":[{"name":"dist_weibull_quantile_complement(1.5, 1.0, 0.05)","align":"left"}],"rows":[["2.078110637534557"]]}}],"relatedNames":["dist_weibull_sample","dist_weibull_pdf","dist_weibull_cdf","dist_weibull_quantile"]},"dist_weibull_range(DOUBLE,DOUBLE)":{"name":"dist_weibull_range","type":"scalar","categories":["Weibull"],"returnType":"DOUBLE[2]","parameters":[{"name":"shape","type":"DOUBLE","paramType":"positional","description":""},{"name":"scale","type":"DOUBLE","paramType":"positional","description":""}],"description":"Returns the range of the weibull distribution.","examples":[{"description":"","code":"SELECT dist_weibull_range(1.5, 1.0);","outputTable":{"columns":[{"name":"dist_weibull_range(1.5, 1.0)","align":"left"}],"rows":[["(0.0, 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support":"dist_cauchy_support(DOUBLE,DOUBLE)","dist_chi_squared_cdf":"dist_chi_squared_cdf(DOUBLE,DOUBLE)","dist_chi_squared_cdf_complement":"dist_chi_squared_cdf_complement(DOUBLE,DOUBLE)","dist_chi_squared_chf":"dist_chi_squared_chf(DOUBLE,DOUBLE)","dist_chi_squared_hazard":"dist_chi_squared_hazard(DOUBLE,DOUBLE)","dist_chi_squared_kurtosis":"dist_chi_squared_kurtosis(DOUBLE)","dist_chi_squared_kurtosis_excess":"dist_chi_squared_kurtosis_excess(DOUBLE)","dist_chi_squared_log_cdf":"dist_chi_squared_log_cdf(DOUBLE,DOUBLE)","dist_chi_squared_log_cdf_complement":"dist_chi_squared_log_cdf_complement(DOUBLE,DOUBLE)","dist_chi_squared_log_pdf":"dist_chi_squared_log_pdf(DOUBLE,DOUBLE)","dist_chi_squared_mean":"dist_chi_squared_mean(DOUBLE)","dist_chi_squared_median":"dist_chi_squared_median(DOUBLE)","dist_chi_squared_mode":"dist_chi_squared_mode(DOUBLE)","dist_chi_squared_pdf":"dist_chi_squared_pdf(DOUBLE,DOUBLE)","dist_chi_squared_quantile":"dist_chi_squared_quantile(DOUBLE,DOUBLE)","dist_chi_squared_quantile_complement":"dist_chi_squared_quantile_complement(DOUBLE,DOUBLE)","dist_chi_squared_range":"dist_chi_squared_range(DOUBLE)","dist_chi_squared_sample":"dist_chi_squared_sample(DOUBLE)","dist_chi_squared_skewness":"dist_chi_squared_skewness(DOUBLE)","dist_chi_squared_stddev":"dist_chi_squared_stddev(DOUBLE)","dist_chi_squared_support":"dist_chi_squared_support(DOUBLE)","dist_chi_squared_variance":"dist_chi_squared_variance(DOUBLE)","dist_exponential_cdf":"dist_exponential_cdf(DOUBLE,DOUBLE)","dist_exponential_cdf_complement":"dist_exponential_cdf_complement(DOUBLE,DOUBLE)","dist_exponential_chf":"dist_exponential_chf(DOUBLE,DOUBLE)","dist_exponential_hazard":"dist_exponential_hazard(DOUBLE,DOUBLE)","dist_exponential_kurtosis":"dist_exponential_kurtosis(DOUBLE)","dist_exponential_kurtosis_excess":"dist_exponential_kurtosis_excess(DOUBLE)","dist_exponential_log_cdf":"dist_exponential_log_cdf(DOUBLE,DOUBLE)","dist_exponential_log_cdf_complement":"dist_exponential_log_cdf_complement(DOUBLE,DOUBLE)","dist_exponential_log_pdf":"dist_exponential_log_pdf(DOUBLE,DOUBLE)","dist_exponential_mean":"dist_exponential_mean(DOUBLE)","dist_exponential_median":"dist_exponential_median(DOUBLE)","dist_exponential_mode":"dist_exponential_mode(DOUBLE)","dist_exponential_pdf":"dist_exponential_pdf(DOUBLE,DOUBLE)","dist_exponential_quantile":"dist_exponential_quantile(DOUBLE,DOUBLE)","dist_exponential_quantile_complement":"dist_exponential_quantile_complement(DOUBLE,DOUBLE)","dist_exponential_range":"dist_exponential_range(DOUBLE)","dist_exponential_sample":"dist_exponential_sample(DOUBLE)","dist_exponential_skewness":"dist_exponential_skewness(DOUBLE)","dist_exponential_stddev":"dist_exponential_stddev(DOUBLE)","dist_exponential_support":"dist_exponential_support(DOUBLE)","dist_exponential_variance":"dist_exponential_variance(DOUBLE)","dist_extreme_value_cdf":"dist_extreme_value_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_extreme_value_cdf_complement":"dist_extreme_value_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_extreme_value_chf":"dist_extreme_value_chf(DOUBLE,DOUBLE,DOUBLE)","dist_extreme_value_hazard":"dist_extreme_value_hazard(DOUBLE,DOUBLE,DOUBLE)","dist_extreme_value_kurtosis":"dist_extreme_value_kurtosis(DOUBLE,DOUBLE)","dist_extreme_value_kurtosis_excess":"dist_extreme_value_kurtosis_excess(DOUBLE,DOUBLE)","dist_extreme_value_log_cdf":"dist_extreme_value_log_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_extreme_value_log_cdf_complement":"dist_extreme_value_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_extreme_value_log_pdf":"dist_extreme_value_log_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_extreme_value_median":"dist_extreme_value_median(DOUBLE,DOUBLE)","dist_extreme_value_mode":"dist_extreme_value_mode(DOUBLE,DOUBLE)","dist_extreme_value_pdf":"dist_extreme_value_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_extreme_value_quantile":"dist_extreme_value_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_extreme_value_quantile_complement":"dist_extreme_value_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_extreme_value_range":"dist_extreme_value_range(DOUBLE,DOUBLE)","dist_extreme_value_sample":"dist_extreme_value_sample(DOUBLE,DOUBLE)","dist_extreme_value_skewness":"dist_extreme_value_skewness(DOUBLE,DOUBLE)","dist_extreme_value_support":"dist_extreme_value_support(DOUBLE,DOUBLE)","dist_extreme_value_variance":"dist_extreme_value_variance(DOUBLE,DOUBLE)","dist_fisher_f_cdf":"dist_fisher_f_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_fisher_f_cdf_complement":"dist_fisher_f_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_fisher_f_chf":"dist_fisher_f_chf(DOUBLE,DOUBLE,DOUBLE)","dist_fisher_f_hazard":"dist_fisher_f_hazard(DOUBLE,DOUBLE,DOUBLE)","dist_fisher_f_kurtosis":"dist_fisher_f_kurtosis(DOUBLE,DOUBLE)","dist_fisher_f_kurtosis_excess":"dist_fisher_f_kurtosis_excess(DOUBLE,DOUBLE)","dist_fisher_f_log_cdf":"dist_fisher_f_log_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_fisher_f_log_cdf_complement":"dist_fisher_f_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_fisher_f_log_pdf":"dist_fisher_f_log_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_fisher_f_median":"dist_fisher_f_median(DOUBLE,DOUBLE)","dist_fisher_f_mode":"dist_fisher_f_mode(DOUBLE,DOUBLE)","dist_fisher_f_pdf":"dist_fisher_f_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_fisher_f_quantile":"dist_fisher_f_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_fisher_f_quantile_complement":"dist_fisher_f_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_fisher_f_range":"dist_fisher_f_range(DOUBLE,DOUBLE)","dist_fisher_f_sample":"dist_fisher_f_sample(DOUBLE,DOUBLE)","dist_fisher_f_skewness":"dist_fisher_f_skewness(DOUBLE,DOUBLE)","dist_fisher_f_support":"dist_fisher_f_support(DOUBLE,DOUBLE)","dist_fisher_f_variance":"dist_fisher_f_variance(DOUBLE,DOUBLE)","dist_gamma_cdf":"dist_gamma_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_gamma_cdf_complement":"dist_gamma_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_gamma_chf":"dist_gamma_chf(DOUBLE,DOUBLE,DOUBLE)","dist_gamma_hazard":"dist_gamma_hazard(DOUBLE,DOUBLE,DOUBLE)","dist_gamma_kurtosis":"dist_gamma_kurtosis(DOUBLE,DOUBLE)","dist_gamma_kurtosis_excess":"dist_gamma_kurtosis_excess(DOUBLE,DOUBLE)","dist_gamma_log_cdf":"dist_gamma_log_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_gamma_log_cdf_complement":"dist_gamma_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_gamma_log_pdf":"dist_gamma_log_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_gamma_mean":"dist_gamma_mean(DOUBLE,DOUBLE)","dist_gamma_median":"dist_gamma_median(DOUBLE,DOUBLE)","dist_gamma_mode":"dist_gamma_mode(DOUBLE,DOUBLE)","dist_gamma_pdf":"dist_gamma_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_gamma_quantile":"dist_gamma_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_gamma_quantile_complement":"dist_gamma_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_gamma_range":"dist_gamma_range(DOUBLE,DOUBLE)","dist_gamma_sample":"dist_gamma_sample(DOUBLE,DOUBLE)","dist_gamma_skewness":"dist_gamma_skewness(DOUBLE,DOUBLE)","dist_gamma_stddev":"dist_gamma_stddev(DOUBLE,DOUBLE)","dist_gamma_support":"dist_gamma_support(DOUBLE,DOUBLE)","dist_gamma_variance":"dist_gamma_variance(DOUBLE,DOUBLE)","dist_geometric_cdf":"dist_geometric_cdf(DOUBLE,DOUBLE)","dist_geometric_cdf_complement":"dist_geometric_cdf_complement(DOUBLE,DOUBLE)","dist_geometric_chf":"dist_geometric_chf(DOUBLE,DOUBLE)","dist_geometric_hazard":"dist_geometric_hazard(DOUBLE,DOUBLE)","dist_geometric_kurtosis":"dist_geometric_kurtosis(DOUBLE)","dist_geometric_kurtosis_excess":"dist_geometric_kurtosis_excess(DOUBLE)","dist_geometric_log_cdf":"dist_geometric_log_cdf(DOUBLE,DOUBLE)","dist_geometric_log_cdf_complement":"dist_geometric_log_cdf_complement(DOUBLE,DOUBLE)","dist_geometric_log_pdf":"dist_geometric_log_pdf(DOUBLE,DOUBLE)","dist_geometric_mean":"dist_geometric_mean(DOUBLE)","dist_geometric_median":"dist_geometric_median(DOUBLE)","dist_geometric_mode":"dist_geometric_mode(DOUBLE)","dist_geometric_pdf":"dist_geometric_pdf(DOUBLE,DOUBLE)","dist_geometric_quantile":"dist_geometric_quantile(DOUBLE,DOUBLE)","dist_geometric_quantile_complement":"dist_geometric_quantile_complement(DOUBLE,DOUBLE)","dist_geometric_range":"dist_geometric_range(DOUBLE)","dist_geometric_sample":"dist_geometric_sample(DOUBLE)","dist_geometric_skewness":"dist_geometric_skewness(DOUBLE)","dist_geometric_stddev":"dist_geometric_stddev(DOUBLE)","dist_geometric_support":"dist_geometric_support(DOUBLE)","dist_geometric_variance":"dist_geometric_variance(DOUBLE)","dist_laplace_cdf":"dist_laplace_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_laplace_cdf_complement":"dist_laplace_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_laplace_chf":"dist_laplace_chf(DOUBLE,DOUBLE,DOUBLE)","dist_laplace_hazard":"dist_laplace_hazard(DOUBLE,DOUBLE,DOUBLE)","dist_laplace_kurtosis":"dist_laplace_kurtosis(DOUBLE,DOUBLE)","dist_laplace_kurtosis_excess":"dist_laplace_kurtosis_excess(DOUBLE,DOUBLE)","dist_laplace_log_cdf":"dist_laplace_log_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_laplace_log_cdf_complement":"dist_laplace_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_laplace_log_pdf":"dist_laplace_log_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_laplace_mean":"dist_laplace_mean(DOUBLE,DOUBLE)","dist_laplace_median":"dist_laplace_median(DOUBLE,DOUBLE)","dist_laplace_mode":"dist_laplace_mode(DOUBLE,DOUBLE)","dist_laplace_pdf":"dist_laplace_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_laplace_quantile":"dist_laplace_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_laplace_quantile_complement":"dist_laplace_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_laplace_range":"dist_laplace_range(DOUBLE,DOUBLE)","dist_laplace_sample":"dist_laplace_sample(DOUBLE,DOUBLE)","dist_laplace_skewness":"dist_laplace_skewness(DOUBLE,DOUBLE)","dist_laplace_stddev":"dist_laplace_stddev(DOUBLE,DOUBLE)","dist_laplace_support":"dist_laplace_support(DOUBLE,DOUBLE)","dist_laplace_variance":"dist_laplace_variance(DOUBLE,DOUBLE)","dist_logistic_cdf":"dist_logistic_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_logistic_cdf_complement":"dist_logistic_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_logistic_chf":"dist_logistic_chf(DOUBLE,DOUBLE,DOUBLE)","dist_logistic_hazard":"dist_logistic_hazard(DOUBLE,DOUBLE,DOUBLE)","dist_logistic_kurtosis":"dist_logistic_kurtosis(DOUBLE,DOUBLE)","dist_logistic_kurtosis_excess":"dist_logistic_kurtosis_excess(DOUBLE,DOUBLE)","dist_logistic_log_cdf":"dist_logistic_log_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_logistic_log_cdf_complement":"dist_logistic_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_logistic_log_pdf":"dist_logistic_log_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_logistic_median":"dist_logistic_median(DOUBLE,DOUBLE)","dist_logistic_mode":"dist_logistic_mode(DOUBLE,DOUBLE)","dist_logistic_pdf":"dist_logistic_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_logistic_quantile":"dist_logistic_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_logistic_quantile_complement":"dist_logistic_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_logistic_range":"dist_logistic_range(DOUBLE,DOUBLE)","dist_logistic_sample":"dist_logistic_sample(DOUBLE,DOUBLE)","dist_logistic_skewness":"dist_logistic_skewness(DOUBLE,DOUBLE)","dist_logistic_support":"dist_logistic_support(DOUBLE,DOUBLE)","dist_logistic_variance":"dist_logistic_variance(DOUBLE,DOUBLE)","dist_lognormal_cdf":"dist_lognormal_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_lognormal_cdf_complement":"dist_lognormal_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_lognormal_chf":"dist_lognormal_chf(DOUBLE,DOUBLE,DOUBLE)","dist_lognormal_hazard":"dist_lognormal_hazard(DOUBLE,DOUBLE,DOUBLE)","dist_lognormal_kurtosis":"dist_lognormal_kurtosis(DOUBLE,DOUBLE)","dist_lognormal_kurtosis_excess":"dist_lognormal_kurtosis_excess(DOUBLE,DOUBLE)","dist_lognormal_log_cdf":"dist_lognormal_log_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_lognormal_log_cdf_complement":"dist_lognormal_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_lognormal_log_pdf":"dist_lognormal_log_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_lognormal_mean":"dist_lognormal_mean(DOUBLE,DOUBLE)","dist_lognormal_median":"dist_lognormal_median(DOUBLE,DOUBLE)","dist_lognormal_mode":"dist_lognormal_mode(DOUBLE,DOUBLE)","dist_lognormal_pdf":"dist_lognormal_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_lognormal_quantile":"dist_lognormal_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_lognormal_quantile_complement":"dist_lognormal_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_lognormal_range":"dist_lognormal_range(DOUBLE,DOUBLE)","dist_lognormal_sample":"dist_lognormal_sample(DOUBLE,DOUBLE)","dist_lognormal_skewness":"dist_lognormal_skewness(DOUBLE,DOUBLE)","dist_lognormal_stddev":"dist_lognormal_stddev(DOUBLE,DOUBLE)","dist_lognormal_support":"dist_lognormal_support(DOUBLE,DOUBLE)","dist_lognormal_variance":"dist_lognormal_variance(DOUBLE,DOUBLE)","dist_negative_binomial_cdf":"dist_negative_binomial_cdf(BIGINT,DOUBLE,DOUBLE)","dist_negative_binomial_cdf_complement":"dist_negative_binomial_cdf_complement(BIGINT,DOUBLE,DOUBLE)","dist_negative_binomial_chf":"dist_negative_binomial_chf(BIGINT,DOUBLE,DOUBLE)","dist_negative_binomial_hazard":"dist_negative_binomial_hazard(BIGINT,DOUBLE,DOUBLE)","dist_negative_binomial_kurtosis":"dist_negative_binomial_kurtosis(BIGINT,DOUBLE)","dist_negative_binomial_kurtosis_excess":"dist_negative_binomial_kurtosis_excess(BIGINT,DOUBLE)","dist_negative_binomial_log_cdf":"dist_negative_binomial_log_cdf(BIGINT,DOUBLE,DOUBLE)","dist_negative_binomial_log_cdf_complement":"dist_negative_binomial_log_cdf_complement(BIGINT,DOUBLE,DOUBLE)","dist_negative_binomial_log_pdf":"dist_negative_binomial_log_pdf(BIGINT,DOUBLE,DOUBLE)","dist_negative_binomial_median":"dist_negative_binomial_median(BIGINT,DOUBLE)","dist_negative_binomial_mode":"dist_negative_binomial_mode(BIGINT,DOUBLE)","dist_negative_binomial_pdf":"dist_negative_binomial_pdf(BIGINT,DOUBLE,DOUBLE)","dist_negative_binomial_quantile":"dist_negative_binomial_quantile(BIGINT,DOUBLE,DOUBLE)","dist_negative_binomial_quantile_complement":"dist_negative_binomial_quantile_complement(BIGINT,DOUBLE,DOUBLE)","dist_negative_binomial_range":"dist_negative_binomial_range(BIGINT,DOUBLE)","dist_negative_binomial_sample":"dist_negative_binomial_sample(BIGINT,DOUBLE)","dist_negative_binomial_skewness":"dist_negative_binomial_skewness(BIGINT,DOUBLE)","dist_negative_binomial_support":"dist_negative_binomial_support(BIGINT,DOUBLE)","dist_negative_binomial_variance":"dist_negative_binomial_variance(BIGINT,DOUBLE)","dist_normal_cdf":"dist_normal_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_normal_cdf_complement":"dist_normal_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_normal_chf":"dist_normal_chf(DOUBLE,DOUBLE,DOUBLE)","dist_normal_hazard":"dist_normal_hazard(DOUBLE,DOUBLE,DOUBLE)","dist_normal_kurtosis":"dist_normal_kurtosis(DOUBLE,DOUBLE)","dist_normal_kurtosis_excess":"dist_normal_kurtosis_excess(DOUBLE,DOUBLE)","dist_normal_log_cdf":"dist_normal_log_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_normal_log_cdf_complement":"dist_normal_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_normal_log_pdf":"dist_normal_log_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_normal_mean":"dist_normal_mean(DOUBLE,DOUBLE)","dist_normal_median":"dist_normal_median(DOUBLE,DOUBLE)","dist_normal_mode":"dist_normal_mode(DOUBLE,DOUBLE)","dist_normal_pdf":"dist_normal_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_normal_quantile":"dist_normal_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_normal_quantile_complement":"dist_normal_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_normal_range":"dist_normal_range(DOUBLE,DOUBLE)","dist_normal_sample":"dist_normal_sample(DOUBLE,DOUBLE)","dist_normal_skewness":"dist_normal_skewness(DOUBLE,DOUBLE)","dist_normal_stddev":"dist_normal_stddev(DOUBLE,DOUBLE)","dist_normal_support":"dist_normal_support(DOUBLE,DOUBLE)","dist_normal_variance":"dist_normal_variance(DOUBLE,DOUBLE)","dist_pareto_cdf":"dist_pareto_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_pareto_cdf_complement":"dist_pareto_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_pareto_chf":"dist_pareto_chf(DOUBLE,DOUBLE,DOUBLE)","dist_pareto_hazard":"dist_pareto_hazard(DOUBLE,DOUBLE,DOUBLE)","dist_pareto_kurtosis":"dist_pareto_kurtosis(DOUBLE,DOUBLE)","dist_pareto_kurtosis_excess":"dist_pareto_kurtosis_excess(DOUBLE,DOUBLE)","dist_pareto_log_cdf":"dist_pareto_log_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_pareto_log_cdf_complement":"dist_pareto_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_pareto_log_pdf":"dist_pareto_log_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_pareto_mean":"dist_pareto_mean(DOUBLE,DOUBLE)","dist_pareto_median":"dist_pareto_median(DOUBLE,DOUBLE)","dist_pareto_mode":"dist_pareto_mode(DOUBLE,DOUBLE)","dist_pareto_pdf":"dist_pareto_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_pareto_quantile":"dist_pareto_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_pareto_quantile_complement":"dist_pareto_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_pareto_range":"dist_pareto_range(DOUBLE,DOUBLE)","dist_pareto_sample":"dist_pareto_sample(DOUBLE,DOUBLE)","dist_pareto_skewness":"dist_pareto_skewness(DOUBLE,DOUBLE)","dist_pareto_stddev":"dist_pareto_stddev(DOUBLE,DOUBLE)","dist_pareto_support":"dist_pareto_support(DOUBLE,DOUBLE)","dist_pareto_variance":"dist_pareto_variance(DOUBLE,DOUBLE)","dist_poisson_cdf":"dist_poisson_cdf(DOUBLE,DOUBLE)","dist_poisson_cdf_complement":"dist_poisson_cdf_complement(DOUBLE,DOUBLE)","dist_poisson_chf":"dist_poisson_chf(DOUBLE,DOUBLE)","dist_poisson_hazard":"dist_poisson_hazard(DOUBLE,DOUBLE)","dist_poisson_kurtosis":"dist_poisson_kurtosis(DOUBLE)","dist_poisson_kurtosis_excess":"dist_poisson_kurtosis_excess(DOUBLE)","dist_poisson_log_cdf":"dist_poisson_log_cdf(DOUBLE,DOUBLE)","dist_poisson_log_cdf_complement":"dist_poisson_log_cdf_complement(DOUBLE,DOUBLE)","dist_poisson_log_pdf":"dist_poisson_log_pdf(DOUBLE,DOUBLE)","dist_poisson_mean":"dist_poisson_mean(DOUBLE)","dist_poisson_median":"dist_poisson_median(DOUBLE)","dist_poisson_mode":"dist_poisson_mode(DOUBLE)","dist_poisson_pdf":"dist_poisson_pdf(DOUBLE,DOUBLE)","dist_poisson_quantile":"dist_poisson_quantile(DOUBLE,DOUBLE)","dist_poisson_quantile_complement":"dist_poisson_quantile_complement(DOUBLE,DOUBLE)","dist_poisson_range":"dist_poisson_range(DOUBLE)","dist_poisson_sample":"dist_poisson_sample(DOUBLE)","dist_poisson_skewness":"dist_poisson_skewness(DOUBLE)","dist_poisson_stddev":"dist_poisson_stddev(DOUBLE)","dist_poisson_support":"dist_poisson_support(DOUBLE)","dist_poisson_variance":"dist_poisson_variance(DOUBLE)","dist_rayleigh_cdf":"dist_rayleigh_cdf(DOUBLE,DOUBLE)","dist_rayleigh_cdf_complement":"dist_rayleigh_cdf_complement(DOUBLE,DOUBLE)","dist_rayleigh_chf":"dist_rayleigh_chf(DOUBLE,DOUBLE)","dist_rayleigh_hazard":"dist_rayleigh_hazard(DOUBLE,DOUBLE)","dist_rayleigh_kurtosis":"dist_rayleigh_kurtosis(DOUBLE)","dist_rayleigh_kurtosis_excess":"dist_rayleigh_kurtosis_excess(DOUBLE)","dist_rayleigh_log_cdf":"dist_rayleigh_log_cdf(DOUBLE,DOUBLE)","dist_rayleigh_log_cdf_complement":"dist_rayleigh_log_cdf_complement(DOUBLE,DOUBLE)","dist_rayleigh_log_pdf":"dist_rayleigh_log_pdf(DOUBLE,DOUBLE)","dist_rayleigh_mean":"dist_rayleigh_mean(DOUBLE)","dist_rayleigh_median":"dist_rayleigh_median(DOUBLE)","dist_rayleigh_mode":"dist_rayleigh_mode(DOUBLE)","dist_rayleigh_pdf":"dist_rayleigh_pdf(DOUBLE,DOUBLE)","dist_rayleigh_quantile":"dist_rayleigh_quantile(DOUBLE,DOUBLE)","dist_rayleigh_quantile_complement":"dist_rayleigh_quantile_complement(DOUBLE,DOUBLE)","dist_rayleigh_range":"dist_rayleigh_range(DOUBLE)","dist_rayleigh_sample":"dist_rayleigh_sample(DOUBLE)","dist_rayleigh_skewness":"dist_rayleigh_skewness(DOUBLE)","dist_rayleigh_stddev":"dist_rayleigh_stddev(DOUBLE)","dist_rayleigh_support":"dist_rayleigh_support(DOUBLE)","dist_rayleigh_variance":"dist_rayleigh_variance(DOUBLE)","dist_students_t_cdf":"dist_students_t_cdf(DOUBLE,DOUBLE)","dist_students_t_cdf_complement":"dist_students_t_cdf_complement(DOUBLE,DOUBLE)","dist_students_t_chf":"dist_students_t_chf(DOUBLE,DOUBLE)","dist_students_t_hazard":"dist_students_t_hazard(DOUBLE,DOUBLE)","dist_students_t_kurtosis":"dist_students_t_kurtosis(DOUBLE)","dist_students_t_kurtosis_excess":"dist_students_t_kurtosis_excess(DOUBLE)","dist_students_t_log_cdf":"dist_students_t_log_cdf(DOUBLE,DOUBLE)","dist_students_t_log_cdf_complement":"dist_students_t_log_cdf_complement(DOUBLE,DOUBLE)","dist_students_t_log_pdf":"dist_students_t_log_pdf(DOUBLE,DOUBLE)","dist_students_t_mean":"dist_students_t_mean(DOUBLE)","dist_students_t_median":"dist_students_t_median(DOUBLE)","dist_students_t_mode":"dist_students_t_mode(DOUBLE)","dist_students_t_pdf":"dist_students_t_pdf(DOUBLE,DOUBLE)","dist_students_t_quantile":"dist_students_t_quantile(DOUBLE,DOUBLE)","dist_students_t_quantile_complement":"dist_students_t_quantile_complement(DOUBLE,DOUBLE)","dist_students_t_range":"dist_students_t_range(DOUBLE)","dist_students_t_sample":"dist_students_t_sample(DOUBLE)","dist_students_t_skewness":"dist_students_t_skewness(DOUBLE)","dist_students_t_stddev":"dist_students_t_stddev(DOUBLE)","dist_students_t_support":"dist_students_t_support(DOUBLE)","dist_students_t_variance":"dist_students_t_variance(DOUBLE)","dist_uniform_int_cdf":"dist_uniform_int_cdf(DOUBLE,DOUBLE,BIGINT)","dist_uniform_int_cdf_complement":"dist_uniform_int_cdf_complement(DOUBLE,DOUBLE,BIGINT)","dist_uniform_int_chf":"dist_uniform_int_chf(DOUBLE,DOUBLE,BIGINT)","dist_uniform_int_hazard":"dist_uniform_int_hazard(DOUBLE,DOUBLE,BIGINT)","dist_uniform_int_kurtosis":"dist_uniform_int_kurtosis(DOUBLE,DOUBLE)","dist_uniform_int_kurtosis_excess":"dist_uniform_int_kurtosis_excess(DOUBLE,DOUBLE)","dist_uniform_int_log_cdf":"dist_uniform_int_log_cdf(DOUBLE,DOUBLE,BIGINT)","dist_uniform_int_log_cdf_complement":"dist_uniform_int_log_cdf_complement(DOUBLE,DOUBLE,BIGINT)","dist_uniform_int_log_pdf":"dist_uniform_int_log_pdf(DOUBLE,DOUBLE,BIGINT)","dist_uniform_int_mean":"dist_uniform_int_mean(DOUBLE,DOUBLE)","dist_uniform_int_median":"dist_uniform_int_median(DOUBLE,DOUBLE)","dist_uniform_int_mode":"dist_uniform_int_mode(DOUBLE,DOUBLE)","dist_uniform_int_pdf":"dist_uniform_int_pdf(DOUBLE,DOUBLE,BIGINT)","dist_uniform_int_quantile":"dist_uniform_int_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_int_quantile_complement":"dist_uniform_int_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_int_range":"dist_uniform_int_range(DOUBLE,DOUBLE)","dist_uniform_int_sample":"dist_uniform_int_sample(DOUBLE,DOUBLE)","dist_uniform_int_skewness":"dist_uniform_int_skewness(DOUBLE,DOUBLE)","dist_uniform_int_stddev":"dist_uniform_int_stddev(DOUBLE,DOUBLE)","dist_uniform_int_support":"dist_uniform_int_support(DOUBLE,DOUBLE)","dist_uniform_int_variance":"dist_uniform_int_variance(DOUBLE,DOUBLE)","dist_uniform_real_cdf":"dist_uniform_real_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_real_cdf_complement":"dist_uniform_real_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_real_chf":"dist_uniform_real_chf(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_real_hazard":"dist_uniform_real_hazard(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_real_kurtosis":"dist_uniform_real_kurtosis(DOUBLE,DOUBLE)","dist_uniform_real_kurtosis_excess":"dist_uniform_real_kurtosis_excess(DOUBLE,DOUBLE)","dist_uniform_real_log_cdf":"dist_uniform_real_log_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_real_log_cdf_complement":"dist_uniform_real_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_real_log_pdf":"dist_uniform_real_log_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_real_mean":"dist_uniform_real_mean(DOUBLE,DOUBLE)","dist_uniform_real_median":"dist_uniform_real_median(DOUBLE,DOUBLE)","dist_uniform_real_mode":"dist_uniform_real_mode(DOUBLE,DOUBLE)","dist_uniform_real_pdf":"dist_uniform_real_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_real_quantile":"dist_uniform_real_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_real_quantile_complement":"dist_uniform_real_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_uniform_real_range":"dist_uniform_real_range(DOUBLE,DOUBLE)","dist_uniform_real_sample":"dist_uniform_real_sample(DOUBLE,DOUBLE)","dist_uniform_real_skewness":"dist_uniform_real_skewness(DOUBLE,DOUBLE)","dist_uniform_real_stddev":"dist_uniform_real_stddev(DOUBLE,DOUBLE)","dist_uniform_real_support":"dist_uniform_real_support(DOUBLE,DOUBLE)","dist_uniform_real_variance":"dist_uniform_real_variance(DOUBLE,DOUBLE)","dist_weibull_cdf":"dist_weibull_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_weibull_cdf_complement":"dist_weibull_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_weibull_chf":"dist_weibull_chf(DOUBLE,DOUBLE,DOUBLE)","dist_weibull_hazard":"dist_weibull_hazard(DOUBLE,DOUBLE,DOUBLE)","dist_weibull_kurtosis":"dist_weibull_kurtosis(DOUBLE,DOUBLE)","dist_weibull_kurtosis_excess":"dist_weibull_kurtosis_excess(DOUBLE,DOUBLE)","dist_weibull_log_cdf":"dist_weibull_log_cdf(DOUBLE,DOUBLE,DOUBLE)","dist_weibull_log_cdf_complement":"dist_weibull_log_cdf_complement(DOUBLE,DOUBLE,DOUBLE)","dist_weibull_log_pdf":"dist_weibull_log_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_weibull_median":"dist_weibull_median(DOUBLE,DOUBLE)","dist_weibull_mode":"dist_weibull_mode(DOUBLE,DOUBLE)","dist_weibull_pdf":"dist_weibull_pdf(DOUBLE,DOUBLE,DOUBLE)","dist_weibull_quantile":"dist_weibull_quantile(DOUBLE,DOUBLE,DOUBLE)","dist_weibull_quantile_complement":"dist_weibull_quantile_complement(DOUBLE,DOUBLE,DOUBLE)","dist_weibull_range":"dist_weibull_range(DOUBLE,DOUBLE)","dist_weibull_sample":"dist_weibull_sample(DOUBLE,DOUBLE)","dist_weibull_skewness":"dist_weibull_skewness(DOUBLE,DOUBLE)","dist_weibull_support":"dist_weibull_support(DOUBLE,DOUBLE)","dist_weibull_variance":"dist_weibull_variance(DOUBLE,DOUBLE)"}}