Stochastic
Monte Carlo, synthetic data, and what-if analysis directly in SQL. 22 named probability distributions, each with the same uniform surface — sample, PDF, CDF, quantile, hazard, plus full distribution properties — for ~444 functions total. Best for ad-hoc statistical work next to warehouse data; for production-grade simulation many users will still want a Python notebook with NumPy and SciPy.
Install
-- Install the extension
INSTALL stochastic FROM community;
-- Load it into your session
LOAD stochastic;
-- Find the 95th percentile of a Normal(0, 1)
SELECT dist_normal_quantile(0.0, 1.0, 0.95) AS p95;
-- 1,000 random draws from N(100, 15)
SELECT dist_normal_sample(100.0, 15.0) AS x
FROM range(1000);
-- P(X ≤ 5 | Binomial(n=10, p=0.3))
SELECT dist_binomial_cdf(10, 0.3, 5) AS prob_at_most_5;Technical Overview
Why Use Stochastic?
Compute probability distributions directly in DuckDB SQL — PDF, CDF, quantile, hazard, and random sampling for 22 named distributions, all under a uniform dist_<family>_<op> naming scheme. Built for Monte Carlo sketches, what-if analysis, and synthetic data generation without leaving SQL.
🎲 What this extension is for
A complete statistical-distributions library exposed as scalar SQL functions. Best fit: ad-hoc analysis, Monte Carlo sketches, synthetic data, p-values, and inline statistical math next to your warehouse data.
- • Monte Carlo and what-if analysis: Combine sampling functions with
range(), aggregate the draws with regular SQL — no Python, no DataFrame round-trip. - • Synthetic data generation: Populate test fixtures or anonymized columns with realistic distributions in a single
CREATE TABLE AS. Mix heights, counts, and wait times from different families in one query. - • Inline statistical math: Compute Z-scores, p-values, Value at Risk, or Bayesian credible intervals as plain SQL expressions wherever a value is allowed.
- • Distribution properties as functions: Mean, median, variance, skewness, kurtosis, support range — every property is a function (not a constant), so it composes inline.
📐 How it works
The extension wraps the Boost.Math statistical-distributions library — the same battle-tested implementations used across C++ statistics tooling. Each of 22 distributions exposes the same 21-operation surface, generated from a single template.
- • Uniform
dist_<family>_<op>naming: Every function is named for its distribution and operation. Operations includesample,pdf,cdf,quantile,hazard, the moments (mean,variance,skewness,kurtosis), and log-scale variants for likelihoods and tail probabilities. - • Vectorized over columns: All functions are scalar — they run column-at-a-time over DuckDB's columnar engine. A million calls inside a
SELECTis one vectorized scan, not a Python loop. - • Strict parameter validation: Illegal parameters raise a SQL error rather than silently returning NaN, so typos in long pipelines fail fast instead of quietly poisoning downstream values.
🛡️ Scope and honest caveats
A deep distributions library, but intentionally not a full simulation framework.
- • Univariate only: No copula or multivariate-normal sampler. For correlated draws, sample independently and apply a Cholesky-style transform yourself.
- • RNG seeding is not user-controlled: There is no documented per-query seed. For exactly-reproducible runs, generate the draws once into a table and treat that table as the seed.
- • No fitting or inference helpers: One-way: parameters → distribution. No MLE / method-of-moments fitters, no built-in tests. Compute the test statistic in SQL and call the appropriate CDF complement for the p-value.
- • Production simulations may want a notebook: For large-scale agent-based simulation, MCMC, or SciPy-grade fitting, keep a Python notebook. Stochastic shines for analyses that live next to warehouse data.
🎯 Common use cases
Monte Carlo over warehouse data
Wrap sampling functions in a CTE over range(N) and aggregate with SQL — estimate π, integrate a payoff function, run a VaR calculation against live position data, all without leaving the warehouse.
Generate realistic synthetic data
Build a fixture table with one CREATE TABLE AS: heights from a Normal, session counts from a Poisson, conversion flags from a Bernoulli. Hundreds of thousands of rows in one query.
Inline p-values and confidence intervals
Compute a Z-statistic in SQL, then call the Normal CDF complement for the two-sided p-value. Same shape works for chi-squared, t, and other tests — analytics dashboards with first-class statistics.
Bayesian conjugate updates
Beta-Binomial, Gamma-Poisson, Normal-Normal — closed-form posteriors are one quantile call away. Useful for A/B-test dashboards and conversion-rate experiments.
Deep Dive
Technical Details
What you can do with one query
Estimate π with a million-draw Monte Carlo — entirely inside DuckDB, no Python, no NumPy:
WITH samples AS (
SELECT dist_uniform_real_sample(-1.0, 1.0) AS x,
dist_uniform_real_sample(-1.0, 1.0) AS y
FROM range(1000000)
)
SELECT 4.0 * COUNT(*) FILTER (WHERE x*x + y*y <= 1) / COUNT(*) AS pi_estimate
FROM samples;The same shape — generate, filter, aggregate — handles portfolio VaR, payoff integration, queueing simulation, or generating a million-row synthetic dataset for a load test. Every dist_<family>_sample is a vectorized scalar function, so a million draws is one columnar scan.
This extension is a statistical-distributions library, not a full simulation framework. It covers the 22 most common distributions with the full PDF / CDF / quantile / hazard / sampling / properties surface — built on the Boost.Math distributions library.
What it deliberately doesn’t ship:
- No copulas / multivariate distributions. The 22 families are univariate; correlated draws need a Cholesky-style transform you write yourself.
- No user-controlled RNG seed. For exactly-reproducible runs, materialize the draws into a table and treat that as the seed.
- No fitters or hypothesis-test wrappers. Compute the test statistic in SQL, then call the appropriate
_cdf_complementfor the p-value.
For production-grade simulation, MCMC, or anything needing SciPy-level fitting, keep a Python notebook with NumPy and SciPy. For quick analysis, what-if exploration, and synthetic data right next to your warehouse data, this is the right tool.
What this extension covers
22 named probability distributions — 16 continuous, 6 discrete — each exposed through the same uniform set of operations:
- Sampling (
_sample) — draw a random value. - Density / mass (
_pdf,_log_pdf) — point density. - Cumulative (
_cdf,_log_cdf,_cdf_complement,_log_cdf_complement) — left tail and survival. - Quantile (
_quantile,_quantile_complement) — inverse CDF. - Hazard (
_hazard,_chf) — instantaneous and cumulative hazard. - Properties (
_mean,_median,_mode,_variance,_stddev,_skewness,_kurtosis,_kurtosis_excess,_range,_support).
The same naming scheme across all 22 means once you know one distribution’s surface you know them all. Browse the categories sidebar to jump straight to the distribution you need.
Why DuckDB instead of Python / R / Excel?
- No data movement. Stats happen where the rows live — no export to NumPy or pandas just to compute a CDF.
- Vectorized. All functions run column-at-a-time over DuckDB’s columnar engine.
- SQL composition. Probability calculations chain cleanly with joins, window functions, CTEs, and CTAS.
- Reproducibility. A
WITHclause containing distribution functions is a complete, self-describing analysis you can paste into any DuckDB session.
Parameter validation
Every function validates its parameters and raises a SQL error on illegal inputs — no silent NaN propagation:
SELECT dist_normal_pdf(0.0, -1.0, 0.5);
-- Error: normal: Standard deviation must be > 0 was: -1.000000
SELECT dist_binomial_pdf(10, 1.5, 5);
-- Error: binomial: Probability must be between 0 and 1 was: 1.500000When to use which family
| Pick this when… | Reach for |
|---|---|
| Test scores, measurement noise, anything CLT-flavoured | Normal |
| Small-sample inference (df ≤ 30) | Student’s t |
| Counts of independent events at a rate | Poisson |
| Yes/no per-trial outcomes | Bernoulli, Binomial |
| Time between events / time-to-failure | Exponential, Weibull |
| Heavy-tailed wealth or file-size data | Pareto, Cauchy, Log-normal |
| Bayesian priors over probabilities | Beta |
| Ratios of variances (ANOVA / regression) | Fisher F |
| Goodness-of-fit, variance tests | Chi-squared |
| Block maxima / rare-event tails | Extreme Value |
| Robust regression / sparse priors | Laplace |
| Inverse-transform sampling, dice, random index | Uniform (Real), Uniform (Integer) |
Reference
Extension Contents
Quick reference to all available functions and settings organized by category.
| Name | Description | |
|---|---|---|
| Bernoulli Bernoulli — single yes/no trial with success probability p. Foundation of every binary outcome model. | ||
| dist_bernoulli_cdf() | Computes the cumulative distribution function (CDF) of the bernoulli distribution | |
| dist_bernoulli_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the bernoulli distribution | |
| dist_bernoulli_chf() | Computes the cumulative hazard function of the bernoulli distribution | |
| dist_bernoulli_hazard() | Computes the hazard function of the bernoulli distribution | |
| dist_bernoulli_kurtosis() | Returns the kurtosis of the bernoulli distribution | |
| dist_bernoulli_kurtosis_excess() | Returns the excess kurtosis of the bernoulli distribution | |
| dist_bernoulli_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the bernoulli distribution | |
| dist_bernoulli_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the bernoulli distribution | |
| dist_bernoulli_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the bernoulli distribution | |
| dist_bernoulli_mean() | Returns the mean (μ) of the bernoulli distribution, which is the first moment | |
| dist_bernoulli_median() | Returns the median (50th percentile) of the bernoulli distribution, which equals the mean | |
| dist_bernoulli_mode() | Returns the mode (most likely value) of the bernoulli distribution, which equals the mean | |
| dist_bernoulli_pdf() | Computes the probability density function (PDF) of the bernoulli distribution | |
| dist_bernoulli_quantile() | Computes the quantile function (inverse CDF) of the bernoulli distribution | |
| dist_bernoulli_quantile_complement() | Computes the complementary quantile function of the bernoulli distribution | |
| dist_bernoulli_range() | Returns the range of the bernoulli distribution | |
| dist_bernoulli_sample() | Generates random samples from the bernoulli distribution with specified parameters | |
| dist_bernoulli_skewness() | Returns the skewness of the bernoulli distribution | |
| dist_bernoulli_stddev() | Returns the standard deviation (σ) of the bernoulli distribution | |
| dist_bernoulli_support() | Returns the support of the bernoulli distribution | |
| dist_bernoulli_variance() | Returns the variance (σ²) of the bernoulli distribution | |
| Beta Beta distribution on [0,1] — shape parameters α, β. Common for Bayesian priors over probabilities and modeling proportions. | ||
| dist_beta_cdf() | Computes the cumulative distribution function (CDF) of the beta distribution | |
| dist_beta_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the beta distribution | |
| dist_beta_chf() | Computes the cumulative hazard function of the beta distribution | |
| dist_beta_hazard() | Computes the hazard function of the beta distribution | |
| dist_beta_kurtosis() | Returns the kurtosis of the beta distribution | |
| dist_beta_kurtosis_excess() | Returns the excess kurtosis of the beta distribution | |
| dist_beta_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the beta distribution | |
| dist_beta_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the beta distribution | |
| dist_beta_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the beta distribution | |
| dist_beta_mean() | Returns the mean (μ) of the beta distribution, which is the first moment | |
| dist_beta_median() | Returns the median (50th percentile) of the beta distribution, which equals the mean | |
| dist_beta_mode() | Returns the mode (most likely value) of the beta distribution, which equals the mean | |
| dist_beta_pdf() | Computes the probability density function (PDF) of the beta distribution | |
| dist_beta_quantile() | Computes the quantile function (inverse CDF) of the beta distribution | |
| dist_beta_quantile_complement() | Computes the complementary quantile function of the beta distribution | |
| dist_beta_range() | Returns the range of the beta distribution | |
| dist_beta_sample() | Generates random samples from the beta distribution with specified parameters | |
| dist_beta_skewness() | Returns the skewness of the beta distribution | |
| dist_beta_stddev() | Returns the standard deviation (σ) of the beta distribution | |
| dist_beta_support() | Returns the support of the beta distribution | |
| dist_beta_variance() | Returns the variance (σ²) of the beta distribution | |
| Binomial Binomial — number of successes in n independent Bernoulli trials at probability p. | ||
| dist_binomial_cdf() | Computes the cumulative distribution function (CDF) of the binomial distribution | |
| dist_binomial_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the binomial distribution | |
| dist_binomial_chf() | Computes the cumulative hazard function of the binomial distribution | |
| dist_binomial_hazard() | Computes the hazard function of the binomial distribution | |
| dist_binomial_kurtosis() | Returns the kurtosis of the binomial distribution | |
| dist_binomial_kurtosis_excess() | Returns the excess kurtosis of the binomial distribution | |
| dist_binomial_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the binomial distribution | |
| dist_binomial_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the binomial distribution | |
| dist_binomial_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the binomial distribution | |
| dist_binomial_median() | Returns the median (50th percentile) of the binomial distribution, which equals the mean | |
| dist_binomial_mode() | Returns the mode (most likely value) of the binomial distribution, which equals the mean | |
| dist_binomial_pdf() | Computes the probability density function (PDF) of the binomial distribution | |
| dist_binomial_quantile() | Computes the quantile function (inverse CDF) of the binomial distribution | |
| dist_binomial_quantile_complement() | Computes the complementary quantile function of the binomial distribution | |
| dist_binomial_range() | Returns the range of the binomial distribution | |
| dist_binomial_sample() | Generates random samples from the binomial distribution with specified parameters | |
| dist_binomial_skewness() | Returns the skewness of the binomial distribution | |
| dist_binomial_support() | Returns the support of the binomial distribution | |
| dist_binomial_variance() | Returns the variance (σ²) of the binomial distribution | |
| Cauchy Cauchy distribution — heavy-tailed location/scale family with no defined mean. Used in robust statistics and physics. | ||
| dist_cauchy_cdf() | Computes the cumulative distribution function (CDF) of the cauchy distribution | |
| dist_cauchy_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the cauchy distribution | |
| dist_cauchy_chf() | Computes the cumulative hazard function of the cauchy distribution | |
| dist_cauchy_hazard() | Computes the hazard function of the cauchy distribution | |
| dist_cauchy_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the cauchy distribution | |
| dist_cauchy_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the cauchy distribution | |
| dist_cauchy_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the cauchy distribution | |
| dist_cauchy_median() | Returns the median (50th percentile) of the cauchy distribution, which equals the mean | |
| dist_cauchy_mode() | Returns the mode (most likely value) of the cauchy distribution, which equals the mean | |
| dist_cauchy_pdf() | Computes the probability density function (PDF) of the cauchy distribution | |
| dist_cauchy_quantile() | Computes the quantile function (inverse CDF) of the cauchy distribution | |
| dist_cauchy_quantile_complement() | Computes the complementary quantile function of the cauchy distribution | |
| dist_cauchy_range() | Returns the range of the cauchy distribution | |
| dist_cauchy_sample() | Generates random samples from the cauchy distribution with specified parameters | |
| dist_cauchy_support() | Returns the support of the cauchy distribution | |
| Chi-squared Chi-squared distribution — sum of k squared standard normals. Used in goodness-of-fit and variance tests. | ||
| dist_chi_squared_cdf() | Computes the cumulative distribution function (CDF) of the chi_squared distribution | |
| dist_chi_squared_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the chi_squared distribution | |
| dist_chi_squared_chf() | Computes the cumulative hazard function of the chi_squared distribution | |
| dist_chi_squared_hazard() | Computes the hazard function of the chi_squared distribution | |
| dist_chi_squared_kurtosis() | Returns the kurtosis of the chi_squared distribution | |
| dist_chi_squared_kurtosis_excess() | Returns the excess kurtosis of the chi_squared distribution | |
| dist_chi_squared_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the chi_squared distribution | |
| dist_chi_squared_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the chi_squared distribution | |
| dist_chi_squared_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the chi_squared distribution | |
| dist_chi_squared_mean() | Returns the mean (μ) of the chi_squared distribution, which is the first moment | |
| dist_chi_squared_median() | Returns the median (50th percentile) of the chi_squared distribution, which equals the mean | |
| dist_chi_squared_mode() | Returns the mode (most likely value) of the chi_squared distribution, which equals the mean | |
| dist_chi_squared_pdf() | Computes the probability density function (PDF) of the chi_squared distribution | |
| dist_chi_squared_quantile() | Computes the quantile function (inverse CDF) of the chi_squared distribution | |
| dist_chi_squared_quantile_complement() | Computes the complementary quantile function of the chi_squared distribution | |
| dist_chi_squared_range() | Returns the range of the chi_squared distribution | |
| dist_chi_squared_sample() | Generates random samples from the chi_squared distribution with specified parameters | |
| dist_chi_squared_skewness() | Returns the skewness of the chi_squared distribution | |
| dist_chi_squared_stddev() | Returns the standard deviation (σ) of the chi_squared distribution | |
| dist_chi_squared_support() | Returns the support of the chi_squared distribution | |
| dist_chi_squared_variance() | Returns the variance (σ²) of the chi_squared distribution | |
| Exponential Exponential distribution — memoryless waiting time between events at rate λ. Survival analysis and reliability. | ||
| dist_exponential_cdf() | Computes the cumulative distribution function (CDF) of the exponential distribution | |
| dist_exponential_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the exponential distribution | |
| dist_exponential_chf() | Computes the cumulative hazard function of the exponential distribution | |
| dist_exponential_hazard() | Computes the hazard function of the exponential distribution | |
| dist_exponential_kurtosis() | Returns the kurtosis of the exponential distribution | |
| dist_exponential_kurtosis_excess() | Returns the excess kurtosis of the exponential distribution | |
| dist_exponential_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the exponential distribution | |
| dist_exponential_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the exponential distribution | |
| dist_exponential_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the exponential distribution | |
| dist_exponential_mean() | Returns the mean (μ) of the exponential distribution, which is the first moment | |
| dist_exponential_median() | Returns the median (50th percentile) of the exponential distribution, which equals the mean | |
| dist_exponential_mode() | Returns the mode (most likely value) of the exponential distribution, which equals the mean | |
| dist_exponential_pdf() | Computes the probability density function (PDF) of the exponential distribution | |
| dist_exponential_quantile() | Computes the quantile function (inverse CDF) of the exponential distribution | |
| dist_exponential_quantile_complement() | Computes the complementary quantile function of the exponential distribution | |
| dist_exponential_range() | Returns the range of the exponential distribution | |
| dist_exponential_sample() | Generates random samples from the exponential distribution with specified parameters | |
| dist_exponential_skewness() | Returns the skewness of the exponential distribution | |
| dist_exponential_stddev() | Returns the standard deviation (σ) of the exponential distribution | |
| dist_exponential_support() | Returns the support of the exponential distribution | |
| dist_exponential_variance() | Returns the variance (σ²) of the exponential distribution | |
| Extreme Value Generalized extreme-value (Gumbel) — block-maxima of i.i.d. samples. Used in hydrology, insurance, climate. | ||
| dist_extreme_value_cdf() | Computes the cumulative distribution function (CDF) of the extreme_value distribution | |
| dist_extreme_value_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the extreme_value distribution | |
| dist_extreme_value_chf() | Computes the cumulative hazard function of the extreme_value distribution | |
| dist_extreme_value_hazard() | Computes the hazard function of the extreme_value distribution | |
| dist_extreme_value_kurtosis() | Returns the kurtosis of the extreme_value distribution | |
| dist_extreme_value_kurtosis_excess() | Returns the excess kurtosis of the extreme_value distribution | |
| dist_extreme_value_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the extreme_value distribution | |
| dist_extreme_value_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the extreme_value distribution | |
| dist_extreme_value_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the extreme_value distribution | |
| dist_extreme_value_median() | Returns the median (50th percentile) of the extreme_value distribution, which equals the mean | |
| dist_extreme_value_mode() | Returns the mode (most likely value) of the extreme_value distribution, which equals the mean | |
| dist_extreme_value_pdf() | Computes the probability density function (PDF) of the extreme_value distribution | |
| dist_extreme_value_quantile() | Computes the quantile function (inverse CDF) of the extreme_value distribution | |
| dist_extreme_value_quantile_complement() | Computes the complementary quantile function of the extreme_value distribution | |
| dist_extreme_value_range() | Returns the range of the extreme_value distribution | |
| dist_extreme_value_sample() | Generates random samples from the extreme_value distribution with specified parameters | |
| dist_extreme_value_skewness() | Returns the skewness of the extreme_value distribution | |
| dist_extreme_value_support() | Returns the support of the extreme_value distribution | |
| dist_extreme_value_variance() | Returns the variance (σ²) of the extreme_value distribution | |
| Fisher F F distribution — ratio of two scaled chi-squared variates. Backbone of ANOVA and regression hypothesis tests. | ||
| dist_fisher_f_cdf() | Computes the cumulative distribution function (CDF) of the fisher_f distribution | |
| dist_fisher_f_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the fisher_f distribution | |
| dist_fisher_f_chf() | Computes the cumulative hazard function of the fisher_f distribution | |
| dist_fisher_f_hazard() | Computes the hazard function of the fisher_f distribution | |
| dist_fisher_f_kurtosis() | Returns the kurtosis of the fisher_f distribution | |
| dist_fisher_f_kurtosis_excess() | Returns the excess kurtosis of the fisher_f distribution | |
| dist_fisher_f_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the fisher_f distribution | |
| dist_fisher_f_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the fisher_f distribution | |
| dist_fisher_f_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the fisher_f distribution | |
| dist_fisher_f_median() | Returns the median (50th percentile) of the fisher_f distribution, which equals the mean | |
| dist_fisher_f_mode() | Returns the mode (most likely value) of the fisher_f distribution, which equals the mean | |
| dist_fisher_f_pdf() | Computes the probability density function (PDF) of the fisher_f distribution | |
| dist_fisher_f_quantile() | Computes the quantile function (inverse CDF) of the fisher_f distribution | |
| dist_fisher_f_quantile_complement() | Computes the complementary quantile function of the fisher_f distribution | |
| dist_fisher_f_range() | Returns the range of the fisher_f distribution | |
| dist_fisher_f_sample() | Generates random samples from the fisher_f distribution with specified parameters | |
| dist_fisher_f_skewness() | Returns the skewness of the fisher_f distribution | |
| dist_fisher_f_support() | Returns the support of the fisher_f distribution | |
| dist_fisher_f_variance() | Returns the variance (σ²) of the fisher_f distribution | |
| Gamma Gamma distribution — shape/scale family. Waiting times for k events at constant rate; flexible right-skewed prior. | ||
| dist_gamma_cdf() | Computes the cumulative distribution function (CDF) of the gamma distribution | |
| dist_gamma_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the gamma distribution | |
| dist_gamma_chf() | Computes the cumulative hazard function of the gamma distribution | |
| dist_gamma_hazard() | Computes the hazard function of the gamma distribution | |
| dist_gamma_kurtosis() | Returns the kurtosis of the gamma distribution | |
| dist_gamma_kurtosis_excess() | Returns the excess kurtosis of the gamma distribution | |
| dist_gamma_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the gamma distribution | |
| dist_gamma_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the gamma distribution | |
| dist_gamma_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the gamma distribution | |
| dist_gamma_mean() | Returns the mean (μ) of the gamma distribution, which is the first moment | |
| dist_gamma_median() | Returns the median (50th percentile) of the gamma distribution, which equals the mean | |
| dist_gamma_mode() | Returns the mode (most likely value) of the gamma distribution, which equals the mean | |
| dist_gamma_pdf() | Computes the probability density function (PDF) of the gamma distribution | |
| dist_gamma_quantile() | Computes the quantile function (inverse CDF) of the gamma distribution | |
| dist_gamma_quantile_complement() | Computes the complementary quantile function of the gamma distribution | |
| dist_gamma_range() | Returns the range of the gamma distribution | |
| dist_gamma_sample() | Generates random samples from the gamma distribution with specified parameters | |
| dist_gamma_skewness() | Returns the skewness of the gamma distribution | |
| dist_gamma_stddev() | Returns the standard deviation (σ) of the gamma distribution | |
| dist_gamma_support() | Returns the support of the gamma distribution | |
| dist_gamma_variance() | Returns the variance (σ²) of the gamma distribution | |
| Geometric Geometric — number of failures before the first success in a Bernoulli trial sequence. | ||
| dist_geometric_cdf() | Computes the cumulative distribution function (CDF) of the geometric distribution | |
| dist_geometric_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the geometric distribution | |
| dist_geometric_chf() | Computes the cumulative hazard function of the geometric distribution | |
| dist_geometric_hazard() | Computes the hazard function of the geometric distribution | |
| dist_geometric_kurtosis() | Returns the kurtosis of the geometric distribution | |
| dist_geometric_kurtosis_excess() | Returns the excess kurtosis of the geometric distribution | |
| dist_geometric_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the geometric distribution | |
| dist_geometric_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the geometric distribution | |
| dist_geometric_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the geometric distribution | |
| dist_geometric_mean() | Returns the mean (μ) of the geometric distribution, which is the first moment | |
| dist_geometric_median() | Returns the median (50th percentile) of the geometric distribution, which equals the mean | |
| dist_geometric_mode() | Returns the mode (most likely value) of the geometric distribution, which equals the mean | |
| dist_geometric_pdf() | Computes the probability density function (PDF) of the geometric distribution | |
| dist_geometric_quantile() | Computes the quantile function (inverse CDF) of the geometric distribution | |
| dist_geometric_quantile_complement() | Computes the complementary quantile function of the geometric distribution | |
| dist_geometric_range() | Returns the range of the geometric distribution | |
| dist_geometric_sample() | Generates random samples from the geometric distribution with specified parameters | |
| dist_geometric_skewness() | Returns the skewness of the geometric distribution | |
| dist_geometric_stddev() | Returns the standard deviation (σ) of the geometric distribution | |
| dist_geometric_support() | Returns the support of the geometric distribution | |
| dist_geometric_variance() | Returns the variance (σ²) of the geometric distribution | |
| Laplace Laplace (double exponential) — location/scale. Heavier tails than Normal; used in regularized regression and noise modeling. | ||
| dist_laplace_cdf() | Computes the cumulative distribution function (CDF) of the laplace distribution | |
| dist_laplace_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the laplace distribution | |
| dist_laplace_chf() | Computes the cumulative hazard function of the laplace distribution | |
| dist_laplace_hazard() | Computes the hazard function of the laplace distribution | |
| dist_laplace_kurtosis() | Returns the kurtosis of the laplace distribution | |
| dist_laplace_kurtosis_excess() | Returns the excess kurtosis of the laplace distribution | |
| dist_laplace_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the laplace distribution | |
| dist_laplace_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the laplace distribution | |
| dist_laplace_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the laplace distribution | |
| dist_laplace_mean() | Returns the mean (μ) of the laplace distribution, which is the first moment | |
| dist_laplace_median() | Returns the median (50th percentile) of the laplace distribution, which equals the mean | |
| dist_laplace_mode() | Returns the mode (most likely value) of the laplace distribution, which equals the mean | |
| dist_laplace_pdf() | Computes the probability density function (PDF) of the laplace distribution | |
| dist_laplace_quantile() | Computes the quantile function (inverse CDF) of the laplace distribution | |
| dist_laplace_quantile_complement() | Computes the complementary quantile function of the laplace distribution | |
| dist_laplace_range() | Returns the range of the laplace distribution | |
| dist_laplace_sample() | Generates random samples from the laplace distribution with specified parameters | |
| dist_laplace_skewness() | Returns the skewness of the laplace distribution | |
| dist_laplace_stddev() | Returns the standard deviation (σ) of the laplace distribution | |
| dist_laplace_support() | Returns the support of the laplace distribution | |
| dist_laplace_variance() | Returns the variance (σ²) of the laplace distribution | |
| Log-normal Log-normal — exp of a Normal. Models multiplicative growth, income, lifetimes; right-skewed with heavy tail. | ||
| dist_lognormal_cdf() | Computes the cumulative distribution function (CDF) of the lognormal distribution | |
| dist_lognormal_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the lognormal distribution | |
| dist_lognormal_chf() | Computes the cumulative hazard function of the lognormal distribution | |
| dist_lognormal_hazard() | Computes the hazard function of the lognormal distribution | |
| dist_lognormal_kurtosis() | Returns the kurtosis of the lognormal distribution | |
| dist_lognormal_kurtosis_excess() | Returns the excess kurtosis of the lognormal distribution | |
| dist_lognormal_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the lognormal distribution | |
| dist_lognormal_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the lognormal distribution | |
| dist_lognormal_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the lognormal distribution | |
| dist_lognormal_mean() | Returns the mean (μ) of the lognormal distribution, which is the first moment | |
| dist_lognormal_median() | Returns the median (50th percentile) of the lognormal distribution, which equals the mean | |
| dist_lognormal_mode() | Returns the mode (most likely value) of the lognormal distribution, which equals the mean | |
| dist_lognormal_pdf() | Computes the probability density function (PDF) of the lognormal distribution | |
| dist_lognormal_quantile() | Computes the quantile function (inverse CDF) of the lognormal distribution | |
| dist_lognormal_quantile_complement() | Computes the complementary quantile function of the lognormal distribution | |
| dist_lognormal_range() | Returns the range of the lognormal distribution | |
| dist_lognormal_sample() | Generates random samples from the lognormal distribution with specified parameters | |
| dist_lognormal_skewness() | Returns the skewness of the lognormal distribution | |
| dist_lognormal_stddev() | Returns the standard deviation (σ) of the lognormal distribution | |
| dist_lognormal_support() | Returns the support of the lognormal distribution | |
| dist_lognormal_variance() | Returns the variance (σ²) of the lognormal distribution | |
| Logistic Logistic distribution — symmetric S-curve via location/scale. Backbone of logistic regression and Elo-style ratings. | ||
| dist_logistic_cdf() | Computes the cumulative distribution function (CDF) of the logistic distribution | |
| dist_logistic_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the logistic distribution | |
| dist_logistic_chf() | Computes the cumulative hazard function of the logistic distribution | |
| dist_logistic_hazard() | Computes the hazard function of the logistic distribution | |
| dist_logistic_kurtosis() | Returns the kurtosis of the logistic distribution | |
| dist_logistic_kurtosis_excess() | Returns the excess kurtosis of the logistic distribution | |
| dist_logistic_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the logistic distribution | |
| dist_logistic_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the logistic distribution | |
| dist_logistic_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the logistic distribution | |
| dist_logistic_median() | Returns the median (50th percentile) of the logistic distribution, which equals the mean | |
| dist_logistic_mode() | Returns the mode (most likely value) of the logistic distribution, which equals the mean | |
| dist_logistic_pdf() | Computes the probability density function (PDF) of the logistic distribution | |
| dist_logistic_quantile() | Computes the quantile function (inverse CDF) of the logistic distribution | |
| dist_logistic_quantile_complement() | Computes the complementary quantile function of the logistic distribution | |
| dist_logistic_range() | Returns the range of the logistic distribution | |
| dist_logistic_sample() | Generates random samples from the logistic distribution with specified parameters | |
| dist_logistic_skewness() | Returns the skewness of the logistic distribution | |
| dist_logistic_support() | Returns the support of the logistic distribution | |
| dist_logistic_variance() | Returns the variance (σ²) of the logistic distribution | |
| Negative Binomial Negative Binomial — number of failures before r successes. Used for over-dispersed count data. | ||
| dist_negative_binomial_cdf() | Computes the cumulative distribution function (CDF) of the negative_binomial distribution | |
| dist_negative_binomial_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the negative_binomial distribution | |
| dist_negative_binomial_chf() | Computes the cumulative hazard function of the negative_binomial distribution | |
| dist_negative_binomial_hazard() | Computes the hazard function of the negative_binomial distribution | |
| dist_negative_binomial_kurtosis() | Returns the kurtosis of the negative_binomial distribution | |
| dist_negative_binomial_kurtosis_excess() | Returns the excess kurtosis of the negative_binomial distribution | |
| dist_negative_binomial_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the negative_binomial distribution | |
| dist_negative_binomial_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the negative_binomial distribution | |
| dist_negative_binomial_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the negative_binomial distribution | |
| dist_negative_binomial_median() | Returns the median (50th percentile) of the negative_binomial distribution, which equals the mean | |
| dist_negative_binomial_mode() | Returns the mode (most likely value) of the negative_binomial distribution, which equals the mean | |
| dist_negative_binomial_pdf() | Computes the probability density function (PDF) of the negative_binomial distribution | |
| dist_negative_binomial_quantile() | Computes the quantile function (inverse CDF) of the negative_binomial distribution | |
| dist_negative_binomial_quantile_complement() | Computes the complementary quantile function of the negative_binomial distribution | |
| dist_negative_binomial_range() | Returns the range of the negative_binomial distribution | |
| dist_negative_binomial_sample() | Generates random samples from the negative_binomial distribution with specified parameters | |
| dist_negative_binomial_skewness() | Returns the skewness of the negative_binomial distribution | |
| dist_negative_binomial_support() | Returns the support of the negative_binomial distribution | |
| dist_negative_binomial_variance() | Returns the variance (σ²) of the negative_binomial distribution | |
| Normal Normal (Gaussian) — the workhorse continuous distribution. Z-scores, confidence intervals, central limit applications. | ||
| dist_normal_cdf() | Computes the cumulative distribution function (CDF) of the normal distribution | |
| dist_normal_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the normal distribution | |
| dist_normal_chf() | Computes the cumulative hazard function of the normal distribution | |
| dist_normal_hazard() | Computes the hazard function of the normal distribution | |
| dist_normal_kurtosis() | Returns the kurtosis of the normal distribution | |
| dist_normal_kurtosis_excess() | Returns the excess kurtosis of the normal distribution | |
| dist_normal_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the normal distribution | |
| dist_normal_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the normal distribution | |
| dist_normal_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the normal distribution | |
| dist_normal_mean() | Returns the mean (μ) of the normal distribution, which is the first moment | |
| dist_normal_median() | Returns the median (50th percentile) of the normal distribution, which equals the mean | |
| dist_normal_mode() | Returns the mode (most likely value) of the normal distribution, which equals the mean | |
| dist_normal_pdf() | Computes the probability density function (PDF) of the normal distribution | |
| dist_normal_quantile() | Computes the quantile function (inverse CDF) of the normal distribution | |
| dist_normal_quantile_complement() | Computes the complementary quantile function of the normal distribution | |
| dist_normal_range() | Returns the range of the normal distribution | |
| dist_normal_sample() | Generates random samples from the normal distribution with specified parameters | |
| dist_normal_skewness() | Returns the skewness of the normal distribution | |
| dist_normal_stddev() | Returns the standard deviation (σ) of the normal distribution | |
| dist_normal_support() | Returns the support of the normal distribution | |
| dist_normal_variance() | Returns the variance (σ²) of the normal distribution | |
| Pareto Pareto distribution — power-law tail with scale and shape. Wealth, file sizes, internet traffic; the 80/20 rule. | ||
| dist_pareto_cdf() | Computes the cumulative distribution function (CDF) of the pareto distribution | |
| dist_pareto_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the pareto distribution | |
| dist_pareto_chf() | Computes the cumulative hazard function of the pareto distribution | |
| dist_pareto_hazard() | Computes the hazard function of the pareto distribution | |
| dist_pareto_kurtosis() | Returns the kurtosis of the pareto distribution | |
| dist_pareto_kurtosis_excess() | Returns the excess kurtosis of the pareto distribution | |
| dist_pareto_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the pareto distribution | |
| dist_pareto_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the pareto distribution | |
| dist_pareto_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the pareto distribution | |
| dist_pareto_mean() | Returns the mean (μ) of the pareto distribution, which is the first moment | |
| dist_pareto_median() | Returns the median (50th percentile) of the pareto distribution, which equals the mean | |
| dist_pareto_mode() | Returns the mode (most likely value) of the pareto distribution, which equals the mean | |
| dist_pareto_pdf() | Computes the probability density function (PDF) of the pareto distribution | |
| dist_pareto_quantile() | Computes the quantile function (inverse CDF) of the pareto distribution | |
| dist_pareto_quantile_complement() | Computes the complementary quantile function of the pareto distribution | |
| dist_pareto_range() | Returns the range of the pareto distribution | |
| dist_pareto_sample() | Generates random samples from the pareto distribution with specified parameters | |
| dist_pareto_skewness() | Returns the skewness of the pareto distribution | |
| dist_pareto_stddev() | Returns the standard deviation (σ) of the pareto distribution | |
| dist_pareto_support() | Returns the support of the pareto distribution | |
| dist_pareto_variance() | Returns the variance (σ²) of the pareto distribution | |
| Poisson Poisson — count of independent events in fixed interval at rate λ. Arrivals, defects, rare-event counts. | ||
| dist_poisson_cdf() | Computes the cumulative distribution function (CDF) of the poisson distribution | |
| dist_poisson_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the poisson distribution | |
| dist_poisson_chf() | Computes the cumulative hazard function of the poisson distribution | |
| dist_poisson_hazard() | Computes the hazard function of the poisson distribution | |
| dist_poisson_kurtosis() | Returns the kurtosis of the poisson distribution | |
| dist_poisson_kurtosis_excess() | Returns the excess kurtosis of the poisson distribution | |
| dist_poisson_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the poisson distribution | |
| dist_poisson_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the poisson distribution | |
| dist_poisson_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the poisson distribution | |
| dist_poisson_mean() | Returns the mean (μ) of the poisson distribution, which is the first moment | |
| dist_poisson_median() | Returns the median (50th percentile) of the poisson distribution, which equals the mean | |
| dist_poisson_mode() | Returns the mode (most likely value) of the poisson distribution, which equals the mean | |
| dist_poisson_pdf() | Computes the probability density function (PDF) of the poisson distribution | |
| dist_poisson_quantile() | Computes the quantile function (inverse CDF) of the poisson distribution | |
| dist_poisson_quantile_complement() | Computes the complementary quantile function of the poisson distribution | |
| dist_poisson_range() | Returns the range of the poisson distribution | |
| dist_poisson_sample() | Generates random samples from the poisson distribution with specified parameters | |
| dist_poisson_skewness() | Returns the skewness of the poisson distribution | |
| dist_poisson_stddev() | Returns the standard deviation (σ) of the poisson distribution | |
| dist_poisson_support() | Returns the support of the poisson distribution | |
| dist_poisson_variance() | Returns the variance (σ²) of the poisson distribution | |
| Rayleigh Rayleigh distribution — magnitude of a 2D vector with i.i.d. Normal components. Wind speed, signal envelopes. | ||
| dist_rayleigh_cdf() | Computes the cumulative distribution function (CDF) of the rayleigh distribution | |
| dist_rayleigh_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the rayleigh distribution | |
| dist_rayleigh_chf() | Computes the cumulative hazard function of the rayleigh distribution | |
| dist_rayleigh_hazard() | Computes the hazard function of the rayleigh distribution | |
| dist_rayleigh_kurtosis() | Returns the kurtosis of the rayleigh distribution | |
| dist_rayleigh_kurtosis_excess() | Returns the excess kurtosis of the rayleigh distribution | |
| dist_rayleigh_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the rayleigh distribution | |
| dist_rayleigh_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the rayleigh distribution | |
| dist_rayleigh_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the rayleigh distribution | |
| dist_rayleigh_mean() | Returns the mean (μ) of the rayleigh distribution, which is the first moment | |
| dist_rayleigh_median() | Returns the median (50th percentile) of the rayleigh distribution, which equals the mean | |
| dist_rayleigh_mode() | Returns the mode (most likely value) of the rayleigh distribution, which equals the mean | |
| dist_rayleigh_pdf() | Computes the probability density function (PDF) of the rayleigh distribution | |
| dist_rayleigh_quantile() | Computes the quantile function (inverse CDF) of the rayleigh distribution | |
| dist_rayleigh_quantile_complement() | Computes the complementary quantile function of the rayleigh distribution | |
| dist_rayleigh_range() | Returns the range of the rayleigh distribution | |
| dist_rayleigh_sample() | Generates random samples from the rayleigh distribution with specified parameters | |
| dist_rayleigh_skewness() | Returns the skewness of the rayleigh distribution | |
| dist_rayleigh_stddev() | Returns the standard deviation (σ) of the rayleigh distribution | |
| dist_rayleigh_support() | Returns the support of the rayleigh distribution | |
| dist_rayleigh_variance() | Returns the variance (σ²) of the rayleigh distribution | |
| Student's t Student's t — heavier-tailed alternative to Normal, parameterized by degrees of freedom. Small-sample inference. | ||
| dist_students_t_cdf() | Computes the cumulative distribution function (CDF) of the students_t distribution | |
| dist_students_t_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the students_t distribution | |
| dist_students_t_chf() | Computes the cumulative hazard function of the students_t distribution | |
| dist_students_t_hazard() | Computes the hazard function of the students_t distribution | |
| dist_students_t_kurtosis() | Returns the kurtosis of the students_t distribution | |
| dist_students_t_kurtosis_excess() | Returns the excess kurtosis of the students_t distribution | |
| dist_students_t_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the students_t distribution | |
| dist_students_t_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the students_t distribution | |
| dist_students_t_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the students_t distribution | |
| dist_students_t_mean() | Returns the mean (μ) of the students_t distribution, which is the first moment | |
| dist_students_t_median() | Returns the median (50th percentile) of the students_t distribution, which equals the mean | |
| dist_students_t_mode() | Returns the mode (most likely value) of the students_t distribution, which equals the mean | |
| dist_students_t_pdf() | Computes the probability density function (PDF) of the students_t distribution | |
| dist_students_t_quantile() | Computes the quantile function (inverse CDF) of the students_t distribution | |
| dist_students_t_quantile_complement() | Computes the complementary quantile function of the students_t distribution | |
| dist_students_t_range() | Returns the range of the students_t distribution | |
| dist_students_t_sample() | Generates random samples from the students_t distribution with specified parameters | |
| dist_students_t_skewness() | Returns the skewness of the students_t distribution | |
| dist_students_t_stddev() | Returns the standard deviation (σ) of the students_t distribution | |
| dist_students_t_support() | Returns the support of the students_t distribution | |
| dist_students_t_variance() | Returns the variance (σ²) of the students_t distribution | |
| Uniform (Integer) Discrete uniform — every integer in [min, max] equally likely. Random index sampling, dice rolls. | ||
| dist_uniform_int_cdf() | Computes the cumulative distribution function (CDF) of the uniform_int distribution | |
| dist_uniform_int_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the uniform_int distribution | |
| dist_uniform_int_chf() | Computes the cumulative hazard function of the uniform_int distribution | |
| dist_uniform_int_hazard() | Computes the hazard function of the uniform_int distribution | |
| dist_uniform_int_kurtosis() | Returns the kurtosis of the uniform_int distribution | |
| dist_uniform_int_kurtosis_excess() | Returns the excess kurtosis of the uniform_int distribution | |
| dist_uniform_int_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the uniform_int distribution | |
| dist_uniform_int_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the uniform_int distribution | |
| dist_uniform_int_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the uniform_int distribution | |
| dist_uniform_int_mean() | Returns the mean (μ) of the uniform_int distribution, which is the first moment | |
| dist_uniform_int_median() | Returns the median (50th percentile) of the uniform_int distribution, which equals the mean | |
| dist_uniform_int_mode() | Returns the mode (most likely value) of the uniform_int distribution, which equals the mean | |
| dist_uniform_int_pdf() | Computes the probability density function (PDF) of the uniform_int distribution | |
| dist_uniform_int_quantile() | Computes the quantile function (inverse CDF) of the uniform_int distribution | |
| dist_uniform_int_quantile_complement() | Computes the complementary quantile function of the uniform_int distribution | |
| dist_uniform_int_range() | Returns the range of the uniform_int distribution | |
| dist_uniform_int_sample() | Generates random samples from the uniform_int distribution with specified parameters | |
| dist_uniform_int_skewness() | Returns the skewness of the uniform_int distribution | |
| dist_uniform_int_stddev() | Returns the standard deviation (σ) of the uniform_int distribution | |
| dist_uniform_int_support() | Returns the support of the uniform_int distribution | |
| dist_uniform_int_variance() | Returns the variance (σ²) of the uniform_int distribution | |
| Uniform (Real) Continuous uniform on [min, max]. The default sampling distribution and a building block for inverse-transform sampling. | ||
| dist_uniform_real_cdf() | Computes the cumulative distribution function (CDF) of the uniform_real distribution | |
| dist_uniform_real_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the uniform_real distribution | |
| dist_uniform_real_chf() | Computes the cumulative hazard function of the uniform_real distribution | |
| dist_uniform_real_hazard() | Computes the hazard function of the uniform_real distribution | |
| dist_uniform_real_kurtosis() | Returns the kurtosis of the uniform_real distribution | |
| dist_uniform_real_kurtosis_excess() | Returns the excess kurtosis of the uniform_real distribution | |
| dist_uniform_real_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the uniform_real distribution | |
| dist_uniform_real_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the uniform_real distribution | |
| dist_uniform_real_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the uniform_real distribution | |
| dist_uniform_real_mean() | Returns the mean (μ) of the uniform_real distribution, which is the first moment | |
| dist_uniform_real_median() | Returns the median (50th percentile) of the uniform_real distribution, which equals the mean | |
| dist_uniform_real_mode() | Returns the mode (most likely value) of the uniform_real distribution, which equals the mean | |
| dist_uniform_real_pdf() | Computes the probability density function (PDF) of the uniform_real distribution | |
| dist_uniform_real_quantile() | Computes the quantile function (inverse CDF) of the uniform_real distribution | |
| dist_uniform_real_quantile_complement() | Computes the complementary quantile function of the uniform_real distribution | |
| dist_uniform_real_range() | Returns the range of the uniform_real distribution | |
| dist_uniform_real_sample() | Generates random samples from the uniform_real distribution with specified parameters | |
| dist_uniform_real_skewness() | Returns the skewness of the uniform_real distribution | |
| dist_uniform_real_stddev() | Returns the standard deviation (σ) of the uniform_real distribution | |
| dist_uniform_real_support() | Returns the support of the uniform_real distribution | |
| dist_uniform_real_variance() | Returns the variance (σ²) of the uniform_real distribution | |
| Weibull Weibull distribution — flexible shape/scale. Reliability engineering, time-to-failure, wind-speed modeling. | ||
| dist_weibull_cdf() | Computes the cumulative distribution function (CDF) of the weibull distribution | |
| dist_weibull_cdf_complement() | Computes the complementary cumulative distribution function (1 - CDF) of the weibull distribution | |
| dist_weibull_chf() | Computes the cumulative hazard function of the weibull distribution | |
| dist_weibull_hazard() | Computes the hazard function of the weibull distribution | |
| dist_weibull_kurtosis() | Returns the kurtosis of the weibull distribution | |
| dist_weibull_kurtosis_excess() | Returns the excess kurtosis of the weibull distribution | |
| dist_weibull_log_cdf() | Computes the natural logarithm of the cumulative distribution function (CDF) of the weibull distribution | |
| dist_weibull_log_cdf_complement() | Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the weibull distribution | |
| dist_weibull_log_pdf() | Computes the natural logarithm of the probability density function (log-PDF) of the weibull distribution | |
| dist_weibull_median() | Returns the median (50th percentile) of the weibull distribution, which equals the mean | |
| dist_weibull_mode() | Returns the mode (most likely value) of the weibull distribution, which equals the mean | |
| dist_weibull_pdf() | Computes the probability density function (PDF) of the weibull distribution | |
| dist_weibull_quantile() | Computes the quantile function (inverse CDF) of the weibull distribution | |
| dist_weibull_quantile_complement() | Computes the complementary quantile function of the weibull distribution | |
| dist_weibull_range() | Returns the range of the weibull distribution | |
| dist_weibull_sample() | Generates random samples from the weibull distribution with specified parameters | |
| dist_weibull_skewness() | Returns the skewness of the weibull distribution | |
| dist_weibull_support() | Returns the support of the weibull distribution | |
| dist_weibull_variance() | Returns the variance (σ²) of the weibull distribution | |
API Reference
Function Documentation
Detailed documentation for each function including signatures, parameters, and examples.
dist_bernoulli_cdf
Signature
Parameters (Positional)
| Parameter | Type | Mode | Description |
|---|---|---|---|
p | DOUBLE | Positional | |
x | DOUBLE | Positional |
Returns
Description
Computes the cumulative distribution function (CDF) of the bernoulli distribution. Returns the probability that a random variable X is less than or equal to x.
Examples
SELECT dist_bernoulli_cdf(0.5, 0);Output
| dist_bernoulli_cdf(0.5, 0) |
|---|
| 0.5 |
Related Functions
- dist_bernoulli_sample() — Generates random samples from the bernoulli distribution with specified parameters
- dist_bernoulli_pdf() — Computes the probability density function (PDF) of the bernoulli distribution
- dist_bernoulli_quantile() — Computes the quantile function (inverse CDF) of the bernoulli distribution
- dist_bernoulli_mean() — Returns the mean (μ) of the bernoulli distribution, which is the first moment
- dist_bernoulli_stddev() — Returns the standard deviation (σ) of the bernoulli distribution
dist_bernoulli_cdf_complement
Signature
Parameters (Positional)
| Parameter | Type | Mode | Description |
|---|---|---|---|
p | DOUBLE | Positional | |
x | DOUBLE | Positional |
Returns
Description
Computes the complementary cumulative distribution function (1 - CDF) of the bernoulli distribution. Returns the probability that X > x, equivalent to the survival function.
Examples
SELECT dist_bernoulli_cdf_complement(0.5, 0);Output
| dist_bernoulli_cdf_complement(0.5, 0) |
|---|
| 0.5 |
Related Functions
- dist_bernoulli_sample() — Generates random samples from the bernoulli distribution with specified parameters
- dist_bernoulli_pdf() — Computes the probability density function (PDF) of the bernoulli distribution
- dist_bernoulli_cdf() — Computes the cumulative distribution function (CDF) of the bernoulli distribution
- dist_bernoulli_quantile() — Computes the quantile function (inverse CDF) of the bernoulli distribution
- dist_bernoulli_mean() — Returns the mean (μ) of the bernoulli distribution, which is the first moment
- dist_bernoulli_stddev() — Returns the standard deviation (σ) of the bernoulli distribution
dist_bernoulli_chf
Signature
Parameters (Positional)
| Parameter | Type | Mode | Description |
|---|---|---|---|
p | DOUBLE | Positional | |
x | DOUBLE | Positional |
Returns
Description
Computes the cumulative hazard function of the bernoulli distribution.
Examples
SELECT dist_bernoulli_chf(0.5, 0);Output
| dist_bernoulli_chf(0.5, 0) |
|---|
| 0.6931471805599453 |
Related Functions
- dist_bernoulli_sample() — Generates random samples from the bernoulli distribution with specified parameters
- dist_bernoulli_pdf() — Computes the probability density function (PDF) of the bernoulli distribution
- dist_bernoulli_cdf() — Computes the cumulative distribution function (CDF) of the bernoulli distribution
- dist_bernoulli_quantile() — Computes the quantile function (inverse CDF) of the bernoulli distribution
- dist_bernoulli_mean() — Returns the mean (μ) of the bernoulli distribution, which is the first moment
- dist_bernoulli_stddev() — Returns the standard deviation (σ) of the bernoulli distribution
dist_bernoulli_hazard
Signature
Parameters (Positional)
| Parameter | Type | Mode | Description |
|---|---|---|---|
p | DOUBLE | Positional | |
x | DOUBLE | Positional |
Returns
Description
Computes the hazard function of the bernoulli distribution.
Examples
SELECT dist_bernoulli_hazard(0.5, 0);Output
| dist_bernoulli_hazard(0.5, 0) |
|---|
| 1.0 |
Related Functions
- dist_bernoulli_sample() — Generates random samples from the bernoulli distribution with specified parameters
- dist_bernoulli_pdf() — Computes the probability density function (PDF) of the bernoulli distribution
- dist_bernoulli_cdf() — Computes the cumulative distribution function (CDF) of the bernoulli distribution
- dist_bernoulli_quantile() — Computes the quantile function (inverse CDF) of the bernoulli distribution
- dist_bernoulli_mean() — Returns the mean (μ) of the bernoulli distribution, which is the first moment
- dist_bernoulli_stddev() — Returns the standard deviation (σ) of the bernoulli distribution
dist_bernoulli_kurtosis
Signature
Parameters (Positional)
| Parameter | Type | Mode | Description |
|---|---|---|---|
p | DOUBLE | Positional |
Returns
Description
Returns the kurtosis of the bernoulli distribution.
Examples
SELECT dist_bernoulli_kurtosis(0.5);Output
| dist_bernoulli_kurtosis(0.5) |
|---|
| 1.0 |
Related Functions
- dist_bernoulli_sample() — Generates random samples from the bernoulli distribution with specified parameters
- dist_bernoulli_pdf() — Computes the probability density function (PDF) of the bernoulli distribution
- dist_bernoulli_cdf() — Computes the cumulative distribution function (CDF) of the bernoulli distribution
- dist_bernoulli_quantile() — Computes the quantile function (inverse CDF) of the bernoulli distribution
- dist_bernoulli_mean() — Returns the mean (μ) of the bernoulli distribution, which is the first moment
- dist_bernoulli_stddev() — Returns the standard deviation (σ) of the bernoulli distribution
dist_bernoulli_kurtosis_excess
Signature
Parameters (Positional)
| Parameter | Type | Mode | Description |
|---|---|---|---|
p | DOUBLE | Positional |
Returns
Description
Returns the excess kurtosis of the bernoulli distribution.
Examples
SELECT dist_bernoulli_kurtosis_excess(0.5);Output
| dist_bernoulli_kurtosis_excess(0.5) |
|---|
| -2.0 |
Related Functions
- dist_bernoulli_sample() — Generates random samples from the bernoulli distribution with specified parameters
- dist_bernoulli_pdf() — Computes the probability density function (PDF) of the bernoulli distribution
- dist_bernoulli_cdf() — Computes the cumulative distribution function (CDF) of the bernoulli distribution
- dist_bernoulli_quantile() — Computes the quantile function (inverse CDF) of the bernoulli distribution
- dist_bernoulli_mean() — Returns the mean (μ) of the bernoulli distribution, which is the first moment
- dist_bernoulli_stddev() — Returns the standard deviation (σ) of the bernoulli distribution
dist_bernoulli_log_cdf
Signature
Parameters (Positional)
| Parameter | Type | Mode | Description |
|---|---|---|---|
p | DOUBLE | Positional | |
x | DOUBLE | Positional |
Returns
Description
Computes the natural logarithm of the cumulative distribution function (CDF) of the bernoulli distribution. Returns the logarithm of the probability that a random variable X is less than or equal to x.
Examples
SELECT dist_bernoulli_log_cdf(0.5, 0);Output
| dist_bernoulli_log_cdf(0.5, 0) |
|---|
| -0.6931471805599453 |
Related Functions
- dist_bernoulli_sample() — Generates random samples from the bernoulli distribution with specified parameters
- dist_bernoulli_pdf() — Computes the probability density function (PDF) of the bernoulli distribution
- dist_bernoulli_cdf() — Computes the cumulative distribution function (CDF) of the bernoulli distribution
- dist_bernoulli_quantile() — Computes the quantile function (inverse CDF) of the bernoulli distribution
- dist_bernoulli_mean() — Returns the mean (μ) of the bernoulli distribution, which is the first moment
- dist_bernoulli_stddev() — Returns the standard deviation (σ) of the bernoulli distribution
dist_bernoulli_log_cdf_complement
Signature
Parameters (Positional)
| Parameter | Type | Mode | Description |
|---|---|---|---|
p | DOUBLE | Positional | |
x | DOUBLE | Positional |
Returns
Description
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the bernoulli distribution. Returns the logarithm of the probability that X > x, equivalent to the survival function.
Examples
SELECT dist_bernoulli_log_cdf_complement(0.5, 0);Output
| dist_bernoulli_log_cdf_complement(0.5, 0) |
|---|
| -0.6931471805599453 |
Related Functions
- dist_bernoulli_sample() — Generates random samples from the bernoulli distribution with specified parameters
- dist_bernoulli_pdf() — Computes the probability density function (PDF) of the bernoulli distribution
- dist_bernoulli_cdf() — Computes the cumulative distribution function (CDF) of the bernoulli distribution
- dist_bernoulli_quantile() — Computes the quantile function (inverse CDF) of the bernoulli distribution
- dist_bernoulli_mean() — Returns the mean (μ) of the bernoulli distribution, which is the first moment
- dist_bernoulli_stddev() — Returns the standard deviation (σ) of the bernoulli distribution
dist_bernoulli_log_pdf
Signature
Parameters (Positional)
| Parameter | Type | Mode | Description |
|---|---|---|---|
p | DOUBLE | Positional | |
x | DOUBLE | Positional |
Returns
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
SELECT dist_bernoulli_log_pdf(0.5, 1);Output
| dist_bernoulli_log_pdf(0.5, 1) |
|---|
| -0.6931471805599453 |
Related Functions
- dist_bernoulli_sample() — Generates random samples from the bernoulli distribution with specified parameters
- dist_bernoulli_pdf() — Computes the probability density function (PDF) of the bernoulli distribution
- dist_bernoulli_cdf() — Computes the cumulative distribution function (CDF) of the bernoulli distribution
- dist_bernoulli_quantile() — Computes the quantile function (inverse CDF) of the bernoulli distribution
- dist_bernoulli_mean() — Returns the mean (μ) of the bernoulli distribution, which is the first moment
- dist_bernoulli_stddev() — Returns the standard deviation (σ) of the bernoulli distribution
dist_bernoulli_mean
Signature
Parameters (Positional)
| Parameter | Type | Mode | Description |
|---|---|---|---|
p | DOUBLE | Positional |
Returns
Description
Returns the mean (μ) of the bernoulli distribution, which is the first moment.
Examples
SELECT dist_bernoulli_mean(0.5);Output
| dist_bernoulli_mean(0.5) |
|---|
| 0.5 |
Related Functions
- dist_bernoulli_sample() — Generates random samples from the bernoulli distribution with specified parameters
- dist_bernoulli_pdf() — Computes the probability density function (PDF) of the bernoulli distribution
- dist_bernoulli_cdf() — Computes the cumulative distribution function (CDF) of the bernoulli distribution
- dist_bernoulli_quantile() — Computes the quantile function (inverse CDF) of the bernoulli distribution
- dist_bernoulli_stddev() — Returns the standard deviation (σ) of the bernoulli distribution
dist_bernoulli_median
Returns the median (50th percentile) of the bernoulli distribution, which equals the mean.
dist_bernoulli_mode
Returns the mode (most likely value) of the bernoulli distribution, which equals the mean.
dist_bernoulli_pdf
Computes the probability density function (PDF) of the bernoulli distribution.
dist_bernoulli_quantile
Computes the quantile function (inverse CDF) of the bernoulli distribution.
dist_bernoulli_quantile_complement
Computes the complementary quantile function of the bernoulli distribution.
dist_bernoulli_range
Returns the range of the bernoulli distribution.
dist_bernoulli_sample
Generates random samples from the bernoulli distribution with specified parameters.
dist_bernoulli_skewness
Returns the skewness of the bernoulli distribution.
dist_bernoulli_stddev
Returns the standard deviation (σ) of the bernoulli distribution.
dist_bernoulli_support
Returns the support of the bernoulli distribution.
dist_bernoulli_variance
Returns the variance (σ²) of the bernoulli distribution.
dist_beta_cdf
Computes the cumulative distribution function (CDF) of the beta distribution.
dist_beta_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the beta distribution.
dist_beta_chf
Computes the cumulative hazard function of the beta distribution.
dist_beta_hazard
Computes the hazard function of the beta distribution.
dist_beta_kurtosis
Returns the kurtosis of the beta distribution.
dist_beta_kurtosis_excess
Returns the excess kurtosis of the beta distribution.
dist_beta_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the beta distribution.
dist_beta_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the beta distribution.
dist_beta_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the beta distribution.
dist_beta_mean
Returns the mean (μ) of the beta distribution, which is the first moment.
dist_beta_median
Returns the median (50th percentile) of the beta distribution, which equals the mean.
dist_beta_mode
Returns the mode (most likely value) of the beta distribution, which equals the mean.
dist_beta_pdf
Computes the probability density function (PDF) of the beta distribution.
dist_beta_quantile
Computes the quantile function (inverse CDF) of the beta distribution.
dist_beta_quantile_complement
Computes the complementary quantile function of the beta distribution.
dist_beta_range
Returns the range of the beta distribution.
dist_beta_sample
Generates random samples from the beta distribution with specified parameters.
dist_beta_skewness
Returns the skewness of the beta distribution.
dist_beta_stddev
Returns the standard deviation (σ) of the beta distribution.
dist_beta_support
Returns the support of the beta distribution.
dist_beta_variance
Returns the variance (σ²) of the beta distribution.
dist_binomial_cdf
Computes the cumulative distribution function (CDF) of the binomial distribution.
dist_binomial_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the binomial distribution.
dist_binomial_chf
Computes the cumulative hazard function of the binomial distribution.
dist_binomial_hazard
Computes the hazard function of the binomial distribution.
dist_binomial_kurtosis
Returns the kurtosis of the binomial distribution.
dist_binomial_kurtosis_excess
Returns the excess kurtosis of the binomial distribution.
dist_binomial_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the binomial distribution.
dist_binomial_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the binomial distribution.
dist_binomial_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the binomial distribution.
dist_binomial_median
Returns the median (50th percentile) of the binomial distribution, which equals the mean.
dist_binomial_mode
Returns the mode (most likely value) of the binomial distribution, which equals the mean.
dist_binomial_pdf
Computes the probability density function (PDF) of the binomial distribution.
dist_binomial_quantile
Computes the quantile function (inverse CDF) of the binomial distribution.
dist_binomial_quantile_complement
Computes the complementary quantile function of the binomial distribution.
dist_binomial_range
Returns the range of the binomial distribution.
dist_binomial_sample
Generates random samples from the binomial distribution with specified parameters.
dist_binomial_skewness
Returns the skewness of the binomial distribution.
dist_binomial_support
Returns the support of the binomial distribution.
dist_binomial_variance
Returns the variance (σ²) of the binomial distribution.
dist_cauchy_cdf
Computes the cumulative distribution function (CDF) of the cauchy distribution.
dist_cauchy_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the cauchy distribution.
dist_cauchy_chf
Computes the cumulative hazard function of the cauchy distribution.
dist_cauchy_hazard
Computes the hazard function of the cauchy distribution.
dist_cauchy_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the cauchy distribution.
dist_cauchy_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the cauchy distribution.
dist_cauchy_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the cauchy distribution.
dist_cauchy_median
Returns the median (50th percentile) of the cauchy distribution, which equals the mean.
dist_cauchy_mode
Returns the mode (most likely value) of the cauchy distribution, which equals the mean.
dist_cauchy_pdf
Computes the probability density function (PDF) of the cauchy distribution.
dist_cauchy_quantile
Computes the quantile function (inverse CDF) of the cauchy distribution.
dist_cauchy_quantile_complement
Computes the complementary quantile function of the cauchy distribution.
dist_cauchy_range
Returns the range of the cauchy distribution.
dist_cauchy_sample
Generates random samples from the cauchy distribution with specified parameters.
dist_cauchy_support
Returns the support of the cauchy distribution.
dist_chi_squared_cdf
Computes the cumulative distribution function (CDF) of the chi_squared distribution.
dist_chi_squared_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the chi_squared distribution.
dist_chi_squared_chf
Computes the cumulative hazard function of the chi_squared distribution.
dist_chi_squared_hazard
Computes the hazard function of the chi_squared distribution.
dist_chi_squared_kurtosis
Returns the kurtosis of the chi_squared distribution.
dist_chi_squared_kurtosis_excess
Returns the excess kurtosis of the chi_squared distribution.
dist_chi_squared_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the chi_squared distribution.
dist_chi_squared_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the chi_squared distribution.
dist_chi_squared_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the chi_squared distribution.
dist_chi_squared_mean
Returns the mean (μ) of the chi_squared distribution, which is the first moment.
dist_chi_squared_median
Returns the median (50th percentile) of the chi_squared distribution, which equals the mean.
dist_chi_squared_mode
Returns the mode (most likely value) of the chi_squared distribution, which equals the mean.
dist_chi_squared_pdf
Computes the probability density function (PDF) of the chi_squared distribution.
dist_chi_squared_quantile
Computes the quantile function (inverse CDF) of the chi_squared distribution.
dist_chi_squared_quantile_complement
Computes the complementary quantile function of the chi_squared distribution.
dist_chi_squared_range
Returns the range of the chi_squared distribution.
dist_chi_squared_sample
Generates random samples from the chi_squared distribution with specified parameters.
dist_chi_squared_skewness
Returns the skewness of the chi_squared distribution.
dist_chi_squared_stddev
Returns the standard deviation (σ) of the chi_squared distribution.
dist_chi_squared_support
Returns the support of the chi_squared distribution.
dist_chi_squared_variance
Returns the variance (σ²) of the chi_squared distribution.
dist_exponential_cdf
Computes the cumulative distribution function (CDF) of the exponential distribution.
dist_exponential_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the exponential distribution.
dist_exponential_chf
Computes the cumulative hazard function of the exponential distribution.
dist_exponential_hazard
Computes the hazard function of the exponential distribution.
dist_exponential_kurtosis
Returns the kurtosis of the exponential distribution.
dist_exponential_kurtosis_excess
Returns the excess kurtosis of the exponential distribution.
dist_exponential_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the exponential distribution.
dist_exponential_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the exponential distribution.
dist_exponential_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the exponential distribution.
dist_exponential_mean
Returns the mean (μ) of the exponential distribution, which is the first moment.
dist_exponential_median
Returns the median (50th percentile) of the exponential distribution, which equals the mean.
dist_exponential_mode
Returns the mode (most likely value) of the exponential distribution, which equals the mean.
dist_exponential_pdf
Computes the probability density function (PDF) of the exponential distribution.
dist_exponential_quantile
Computes the quantile function (inverse CDF) of the exponential distribution.
dist_exponential_quantile_complement
Computes the complementary quantile function of the exponential distribution.
dist_exponential_range
Returns the range of the exponential distribution.
dist_exponential_sample
Generates random samples from the exponential distribution with specified parameters.
dist_exponential_skewness
Returns the skewness of the exponential distribution.
dist_exponential_stddev
Returns the standard deviation (σ) of the exponential distribution.
dist_exponential_support
Returns the support of the exponential distribution.
dist_exponential_variance
Returns the variance (σ²) of the exponential distribution.
dist_extreme_value_cdf
Computes the cumulative distribution function (CDF) of the extreme_value distribution.
dist_extreme_value_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the extreme_value distribution.
dist_extreme_value_chf
Computes the cumulative hazard function of the extreme_value distribution.
dist_extreme_value_hazard
Computes the hazard function of the extreme_value distribution.
dist_extreme_value_kurtosis
Returns the kurtosis of the extreme_value distribution.
dist_extreme_value_kurtosis_excess
Returns the excess kurtosis of the extreme_value distribution.
dist_extreme_value_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the extreme_value distribution.
dist_extreme_value_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the extreme_value distribution.
dist_extreme_value_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the extreme_value distribution.
dist_extreme_value_median
Returns the median (50th percentile) of the extreme_value distribution, which equals the mean.
dist_extreme_value_mode
Returns the mode (most likely value) of the extreme_value distribution, which equals the mean.
dist_extreme_value_pdf
Computes the probability density function (PDF) of the extreme_value distribution.
dist_extreme_value_quantile
Computes the quantile function (inverse CDF) of the extreme_value distribution.
dist_extreme_value_quantile_complement
Computes the complementary quantile function of the extreme_value distribution.
dist_extreme_value_range
Returns the range of the extreme_value distribution.
dist_extreme_value_sample
Generates random samples from the extreme_value distribution with specified parameters.
dist_extreme_value_skewness
Returns the skewness of the extreme_value distribution.
dist_extreme_value_support
Returns the support of the extreme_value distribution.
dist_extreme_value_variance
Returns the variance (σ²) of the extreme_value distribution.
dist_fisher_f_cdf
Computes the cumulative distribution function (CDF) of the fisher_f distribution.
dist_fisher_f_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the fisher_f distribution.
dist_fisher_f_chf
Computes the cumulative hazard function of the fisher_f distribution.
dist_fisher_f_hazard
Computes the hazard function of the fisher_f distribution.
dist_fisher_f_kurtosis
Returns the kurtosis of the fisher_f distribution.
dist_fisher_f_kurtosis_excess
Returns the excess kurtosis of the fisher_f distribution.
dist_fisher_f_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the fisher_f distribution.
dist_fisher_f_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the fisher_f distribution.
dist_fisher_f_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the fisher_f distribution.
dist_fisher_f_median
Returns the median (50th percentile) of the fisher_f distribution, which equals the mean.
dist_fisher_f_mode
Returns the mode (most likely value) of the fisher_f distribution, which equals the mean.
dist_fisher_f_pdf
Computes the probability density function (PDF) of the fisher_f distribution.
dist_fisher_f_quantile
Computes the quantile function (inverse CDF) of the fisher_f distribution.
dist_fisher_f_quantile_complement
Computes the complementary quantile function of the fisher_f distribution.
dist_fisher_f_range
Returns the range of the fisher_f distribution.
dist_fisher_f_sample
Generates random samples from the fisher_f distribution with specified parameters.
dist_fisher_f_skewness
Returns the skewness of the fisher_f distribution.
dist_fisher_f_support
Returns the support of the fisher_f distribution.
dist_fisher_f_variance
Returns the variance (σ²) of the fisher_f distribution.
dist_gamma_cdf
Computes the cumulative distribution function (CDF) of the gamma distribution.
dist_gamma_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the gamma distribution.
dist_gamma_chf
Computes the cumulative hazard function of the gamma distribution.
dist_gamma_hazard
Computes the hazard function of the gamma distribution.
dist_gamma_kurtosis
Returns the kurtosis of the gamma distribution.
dist_gamma_kurtosis_excess
Returns the excess kurtosis of the gamma distribution.
dist_gamma_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the gamma distribution.
dist_gamma_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the gamma distribution.
dist_gamma_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the gamma distribution.
dist_gamma_mean
Returns the mean (μ) of the gamma distribution, which is the first moment.
dist_gamma_median
Returns the median (50th percentile) of the gamma distribution, which equals the mean.
dist_gamma_mode
Returns the mode (most likely value) of the gamma distribution, which equals the mean.
dist_gamma_pdf
Computes the probability density function (PDF) of the gamma distribution.
dist_gamma_quantile
Computes the quantile function (inverse CDF) of the gamma distribution.
dist_gamma_quantile_complement
Computes the complementary quantile function of the gamma distribution.
dist_gamma_range
Returns the range of the gamma distribution.
dist_gamma_sample
Generates random samples from the gamma distribution with specified parameters.
dist_gamma_skewness
Returns the skewness of the gamma distribution.
dist_gamma_stddev
Returns the standard deviation (σ) of the gamma distribution.
dist_gamma_support
Returns the support of the gamma distribution.
dist_gamma_variance
Returns the variance (σ²) of the gamma distribution.
dist_geometric_cdf
Computes the cumulative distribution function (CDF) of the geometric distribution.
dist_geometric_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the geometric distribution.
dist_geometric_chf
Computes the cumulative hazard function of the geometric distribution.
dist_geometric_hazard
Computes the hazard function of the geometric distribution.
dist_geometric_kurtosis
Returns the kurtosis of the geometric distribution.
dist_geometric_kurtosis_excess
Returns the excess kurtosis of the geometric distribution.
dist_geometric_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the geometric distribution.
dist_geometric_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the geometric distribution.
dist_geometric_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the geometric distribution.
dist_geometric_mean
Returns the mean (μ) of the geometric distribution, which is the first moment.
dist_geometric_median
Returns the median (50th percentile) of the geometric distribution, which equals the mean.
dist_geometric_mode
Returns the mode (most likely value) of the geometric distribution, which equals the mean.
dist_geometric_pdf
Computes the probability density function (PDF) of the geometric distribution.
dist_geometric_quantile
Computes the quantile function (inverse CDF) of the geometric distribution.
dist_geometric_quantile_complement
Computes the complementary quantile function of the geometric distribution.
dist_geometric_range
Returns the range of the geometric distribution.
dist_geometric_sample
Generates random samples from the geometric distribution with specified parameters.
dist_geometric_skewness
Returns the skewness of the geometric distribution.
dist_geometric_stddev
Returns the standard deviation (σ) of the geometric distribution.
dist_geometric_support
Returns the support of the geometric distribution.
dist_geometric_variance
Returns the variance (σ²) of the geometric distribution.
dist_laplace_cdf
Computes the cumulative distribution function (CDF) of the laplace distribution.
dist_laplace_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the laplace distribution.
dist_laplace_chf
Computes the cumulative hazard function of the laplace distribution.
dist_laplace_hazard
Computes the hazard function of the laplace distribution.
dist_laplace_kurtosis
Returns the kurtosis of the laplace distribution.
dist_laplace_kurtosis_excess
Returns the excess kurtosis of the laplace distribution.
dist_laplace_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the laplace distribution.
dist_laplace_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the laplace distribution.
dist_laplace_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the laplace distribution.
dist_laplace_mean
Returns the mean (μ) of the laplace distribution, which is the first moment.
dist_laplace_median
Returns the median (50th percentile) of the laplace distribution, which equals the mean.
dist_laplace_mode
Returns the mode (most likely value) of the laplace distribution, which equals the mean.
dist_laplace_pdf
Computes the probability density function (PDF) of the laplace distribution.
dist_laplace_quantile
Computes the quantile function (inverse CDF) of the laplace distribution.
dist_laplace_quantile_complement
Computes the complementary quantile function of the laplace distribution.
dist_laplace_range
Returns the range of the laplace distribution.
dist_laplace_sample
Generates random samples from the laplace distribution with specified parameters.
dist_laplace_skewness
Returns the skewness of the laplace distribution.
dist_laplace_stddev
Returns the standard deviation (σ) of the laplace distribution.
dist_laplace_support
Returns the support of the laplace distribution.
dist_laplace_variance
Returns the variance (σ²) of the laplace distribution.
dist_logistic_cdf
Computes the cumulative distribution function (CDF) of the logistic distribution.
dist_logistic_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the logistic distribution.
dist_logistic_chf
Computes the cumulative hazard function of the logistic distribution.
dist_logistic_hazard
Computes the hazard function of the logistic distribution.
dist_logistic_kurtosis
Returns the kurtosis of the logistic distribution.
dist_logistic_kurtosis_excess
Returns the excess kurtosis of the logistic distribution.
dist_logistic_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the logistic distribution.
dist_logistic_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the logistic distribution.
dist_logistic_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the logistic distribution.
dist_logistic_median
Returns the median (50th percentile) of the logistic distribution, which equals the mean.
dist_logistic_mode
Returns the mode (most likely value) of the logistic distribution, which equals the mean.
dist_logistic_pdf
Computes the probability density function (PDF) of the logistic distribution.
dist_logistic_quantile
Computes the quantile function (inverse CDF) of the logistic distribution.
dist_logistic_quantile_complement
Computes the complementary quantile function of the logistic distribution.
dist_logistic_range
Returns the range of the logistic distribution.
dist_logistic_sample
Generates random samples from the logistic distribution with specified parameters.
dist_logistic_skewness
Returns the skewness of the logistic distribution.
dist_logistic_support
Returns the support of the logistic distribution.
dist_logistic_variance
Returns the variance (σ²) of the logistic distribution.
dist_lognormal_cdf
Computes the cumulative distribution function (CDF) of the lognormal distribution.
dist_lognormal_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the lognormal distribution.
dist_lognormal_chf
Computes the cumulative hazard function of the lognormal distribution.
dist_lognormal_hazard
Computes the hazard function of the lognormal distribution.
dist_lognormal_kurtosis
Returns the kurtosis of the lognormal distribution.
dist_lognormal_kurtosis_excess
Returns the excess kurtosis of the lognormal distribution.
dist_lognormal_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the lognormal distribution.
dist_lognormal_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the lognormal distribution.
dist_lognormal_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the lognormal distribution.
dist_lognormal_mean
Returns the mean (μ) of the lognormal distribution, which is the first moment.
dist_lognormal_median
Returns the median (50th percentile) of the lognormal distribution, which equals the mean.
dist_lognormal_mode
Returns the mode (most likely value) of the lognormal distribution, which equals the mean.
dist_lognormal_pdf
Computes the probability density function (PDF) of the lognormal distribution.
dist_lognormal_quantile
Computes the quantile function (inverse CDF) of the lognormal distribution.
dist_lognormal_quantile_complement
Computes the complementary quantile function of the lognormal distribution.
dist_lognormal_range
Returns the range of the lognormal distribution.
dist_lognormal_sample
Generates random samples from the lognormal distribution with specified parameters.
dist_lognormal_skewness
Returns the skewness of the lognormal distribution.
dist_lognormal_stddev
Returns the standard deviation (σ) of the lognormal distribution.
dist_lognormal_support
Returns the support of the lognormal distribution.
dist_lognormal_variance
Returns the variance (σ²) of the lognormal distribution.
dist_negative_binomial_cdf
Computes the cumulative distribution function (CDF) of the negative_binomial distribution.
dist_negative_binomial_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the negative_binomial distribution.
dist_negative_binomial_chf
Computes the cumulative hazard function of the negative_binomial distribution.
dist_negative_binomial_hazard
Computes the hazard function of the negative_binomial distribution.
dist_negative_binomial_kurtosis
Returns the kurtosis of the negative_binomial distribution.
dist_negative_binomial_kurtosis_excess
Returns the excess kurtosis of the negative_binomial distribution.
dist_negative_binomial_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the negative_binomial distribution.
dist_negative_binomial_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the negative_binomial distribution.
dist_negative_binomial_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the negative_binomial distribution.
dist_negative_binomial_median
Returns the median (50th percentile) of the negative_binomial distribution, which equals the mean.
dist_negative_binomial_mode
Returns the mode (most likely value) of the negative_binomial distribution, which equals the mean.
dist_negative_binomial_pdf
Computes the probability density function (PDF) of the negative_binomial distribution.
dist_negative_binomial_quantile
Computes the quantile function (inverse CDF) of the negative_binomial distribution.
dist_negative_binomial_quantile_complement
Computes the complementary quantile function of the negative_binomial distribution.
dist_negative_binomial_range
Returns the range of the negative_binomial distribution.
dist_negative_binomial_sample
Generates random samples from the negative_binomial distribution with specified parameters.
dist_negative_binomial_skewness
Returns the skewness of the negative_binomial distribution.
dist_negative_binomial_support
Returns the support of the negative_binomial distribution.
dist_negative_binomial_variance
Returns the variance (σ²) of the negative_binomial distribution.
dist_normal_cdf
Computes the cumulative distribution function (CDF) of the normal distribution.
dist_normal_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the normal distribution.
dist_normal_chf
Computes the cumulative hazard function of the normal distribution.
dist_normal_hazard
Computes the hazard function of the normal distribution.
dist_normal_kurtosis
Returns the kurtosis of the normal distribution.
dist_normal_kurtosis_excess
Returns the excess kurtosis of the normal distribution.
dist_normal_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the normal distribution.
dist_normal_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the normal distribution.
dist_normal_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the normal distribution.
dist_normal_mean
Returns the mean (μ) of the normal distribution, which is the first moment.
dist_normal_median
Returns the median (50th percentile) of the normal distribution, which equals the mean.
dist_normal_mode
Returns the mode (most likely value) of the normal distribution, which equals the mean.
dist_normal_pdf
Computes the probability density function (PDF) of the normal distribution.
dist_normal_quantile
Computes the quantile function (inverse CDF) of the normal distribution.
dist_normal_quantile_complement
Computes the complementary quantile function of the normal distribution.
dist_normal_range
Returns the range of the normal distribution.
dist_normal_sample
Generates random samples from the normal distribution with specified parameters.
dist_normal_skewness
Returns the skewness of the normal distribution.
dist_normal_stddev
Returns the standard deviation (σ) of the normal distribution.
dist_normal_support
Returns the support of the normal distribution.
dist_normal_variance
Returns the variance (σ²) of the normal distribution.
dist_pareto_cdf
Computes the cumulative distribution function (CDF) of the pareto distribution.
dist_pareto_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the pareto distribution.
dist_pareto_chf
Computes the cumulative hazard function of the pareto distribution.
dist_pareto_hazard
Computes the hazard function of the pareto distribution.
dist_pareto_kurtosis
Returns the kurtosis of the pareto distribution.
dist_pareto_kurtosis_excess
Returns the excess kurtosis of the pareto distribution.
dist_pareto_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the pareto distribution.
dist_pareto_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the pareto distribution.
dist_pareto_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the pareto distribution.
dist_pareto_mean
Returns the mean (μ) of the pareto distribution, which is the first moment.
dist_pareto_median
Returns the median (50th percentile) of the pareto distribution, which equals the mean.
dist_pareto_mode
Returns the mode (most likely value) of the pareto distribution, which equals the mean.
dist_pareto_pdf
Computes the probability density function (PDF) of the pareto distribution.
dist_pareto_quantile
Computes the quantile function (inverse CDF) of the pareto distribution.
dist_pareto_quantile_complement
Computes the complementary quantile function of the pareto distribution.
dist_pareto_range
Returns the range of the pareto distribution.
dist_pareto_sample
Generates random samples from the pareto distribution with specified parameters.
dist_pareto_skewness
Returns the skewness of the pareto distribution.
dist_pareto_stddev
Returns the standard deviation (σ) of the pareto distribution.
dist_pareto_support
Returns the support of the pareto distribution.
dist_pareto_variance
Returns the variance (σ²) of the pareto distribution.
dist_poisson_cdf
Computes the cumulative distribution function (CDF) of the poisson distribution.
dist_poisson_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the poisson distribution.
dist_poisson_chf
Computes the cumulative hazard function of the poisson distribution.
dist_poisson_hazard
Computes the hazard function of the poisson distribution.
dist_poisson_kurtosis
Returns the kurtosis of the poisson distribution.
dist_poisson_kurtosis_excess
Returns the excess kurtosis of the poisson distribution.
dist_poisson_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the poisson distribution.
dist_poisson_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the poisson distribution.
dist_poisson_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the poisson distribution.
dist_poisson_mean
Returns the mean (μ) of the poisson distribution, which is the first moment.
dist_poisson_median
Returns the median (50th percentile) of the poisson distribution, which equals the mean.
dist_poisson_mode
Returns the mode (most likely value) of the poisson distribution, which equals the mean.
dist_poisson_pdf
Computes the probability density function (PDF) of the poisson distribution.
dist_poisson_quantile
Computes the quantile function (inverse CDF) of the poisson distribution.
dist_poisson_quantile_complement
Computes the complementary quantile function of the poisson distribution.
dist_poisson_range
Returns the range of the poisson distribution.
dist_poisson_sample
Generates random samples from the poisson distribution with specified parameters.
dist_poisson_skewness
Returns the skewness of the poisson distribution.
dist_poisson_stddev
Returns the standard deviation (σ) of the poisson distribution.
dist_poisson_support
Returns the support of the poisson distribution.
dist_poisson_variance
Returns the variance (σ²) of the poisson distribution.
dist_rayleigh_cdf
Computes the cumulative distribution function (CDF) of the rayleigh distribution.
dist_rayleigh_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the rayleigh distribution.
dist_rayleigh_chf
Computes the cumulative hazard function of the rayleigh distribution.
dist_rayleigh_hazard
Computes the hazard function of the rayleigh distribution.
dist_rayleigh_kurtosis
Returns the kurtosis of the rayleigh distribution.
dist_rayleigh_kurtosis_excess
Returns the excess kurtosis of the rayleigh distribution.
dist_rayleigh_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the rayleigh distribution.
dist_rayleigh_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the rayleigh distribution.
dist_rayleigh_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the rayleigh distribution.
dist_rayleigh_mean
Returns the mean (μ) of the rayleigh distribution, which is the first moment.
dist_rayleigh_median
Returns the median (50th percentile) of the rayleigh distribution, which equals the mean.
dist_rayleigh_mode
Returns the mode (most likely value) of the rayleigh distribution, which equals the mean.
dist_rayleigh_pdf
Computes the probability density function (PDF) of the rayleigh distribution.
dist_rayleigh_quantile
Computes the quantile function (inverse CDF) of the rayleigh distribution.
dist_rayleigh_quantile_complement
Computes the complementary quantile function of the rayleigh distribution.
dist_rayleigh_range
Returns the range of the rayleigh distribution.
dist_rayleigh_sample
Generates random samples from the rayleigh distribution with specified parameters.
dist_rayleigh_skewness
Returns the skewness of the rayleigh distribution.
dist_rayleigh_stddev
Returns the standard deviation (σ) of the rayleigh distribution.
dist_rayleigh_support
Returns the support of the rayleigh distribution.
dist_rayleigh_variance
Returns the variance (σ²) of the rayleigh distribution.
dist_students_t_cdf
Computes the cumulative distribution function (CDF) of the students_t distribution.
dist_students_t_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the students_t distribution.
dist_students_t_chf
Computes the cumulative hazard function of the students_t distribution.
dist_students_t_hazard
Computes the hazard function of the students_t distribution.
dist_students_t_kurtosis
Returns the kurtosis of the students_t distribution.
dist_students_t_kurtosis_excess
Returns the excess kurtosis of the students_t distribution.
dist_students_t_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the students_t distribution.
dist_students_t_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the students_t distribution.
dist_students_t_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the students_t distribution.
dist_students_t_mean
Returns the mean (μ) of the students_t distribution, which is the first moment.
dist_students_t_median
Returns the median (50th percentile) of the students_t distribution, which equals the mean.
dist_students_t_mode
Returns the mode (most likely value) of the students_t distribution, which equals the mean.
dist_students_t_pdf
Computes the probability density function (PDF) of the students_t distribution.
dist_students_t_quantile
Computes the quantile function (inverse CDF) of the students_t distribution.
dist_students_t_quantile_complement
Computes the complementary quantile function of the students_t distribution.
dist_students_t_range
Returns the range of the students_t distribution.
dist_students_t_sample
Generates random samples from the students_t distribution with specified parameters.
dist_students_t_skewness
Returns the skewness of the students_t distribution.
dist_students_t_stddev
Returns the standard deviation (σ) of the students_t distribution.
dist_students_t_support
Returns the support of the students_t distribution.
dist_students_t_variance
Returns the variance (σ²) of the students_t distribution.
dist_uniform_int_cdf
Computes the cumulative distribution function (CDF) of the uniform_int distribution.
dist_uniform_int_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the uniform_int distribution.
dist_uniform_int_chf
Computes the cumulative hazard function of the uniform_int distribution.
dist_uniform_int_hazard
Computes the hazard function of the uniform_int distribution.
dist_uniform_int_kurtosis
Returns the kurtosis of the uniform_int distribution.
dist_uniform_int_kurtosis_excess
Returns the excess kurtosis of the uniform_int distribution.
dist_uniform_int_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the uniform_int distribution.
dist_uniform_int_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the uniform_int distribution.
dist_uniform_int_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the uniform_int distribution.
dist_uniform_int_mean
Returns the mean (μ) of the uniform_int distribution, which is the first moment.
dist_uniform_int_median
Returns the median (50th percentile) of the uniform_int distribution, which equals the mean.
dist_uniform_int_mode
Returns the mode (most likely value) of the uniform_int distribution, which equals the mean.
dist_uniform_int_pdf
Computes the probability density function (PDF) of the uniform_int distribution.
dist_uniform_int_quantile
Computes the quantile function (inverse CDF) of the uniform_int distribution.
dist_uniform_int_quantile_complement
Computes the complementary quantile function of the uniform_int distribution.
dist_uniform_int_range
Returns the range of the uniform_int distribution.
dist_uniform_int_sample
Generates random samples from the uniform_int distribution with specified parameters.
dist_uniform_int_skewness
Returns the skewness of the uniform_int distribution.
dist_uniform_int_stddev
Returns the standard deviation (σ) of the uniform_int distribution.
dist_uniform_int_support
Returns the support of the uniform_int distribution.
dist_uniform_int_variance
Returns the variance (σ²) of the uniform_int distribution.
dist_uniform_real_cdf
Computes the cumulative distribution function (CDF) of the uniform_real distribution.
dist_uniform_real_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the uniform_real distribution.
dist_uniform_real_chf
Computes the cumulative hazard function of the uniform_real distribution.
dist_uniform_real_hazard
Computes the hazard function of the uniform_real distribution.
dist_uniform_real_kurtosis
Returns the kurtosis of the uniform_real distribution.
dist_uniform_real_kurtosis_excess
Returns the excess kurtosis of the uniform_real distribution.
dist_uniform_real_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the uniform_real distribution.
dist_uniform_real_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the uniform_real distribution.
dist_uniform_real_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the uniform_real distribution.
dist_uniform_real_mean
Returns the mean (μ) of the uniform_real distribution, which is the first moment.
dist_uniform_real_median
Returns the median (50th percentile) of the uniform_real distribution, which equals the mean.
dist_uniform_real_mode
Returns the mode (most likely value) of the uniform_real distribution, which equals the mean.
dist_uniform_real_pdf
Computes the probability density function (PDF) of the uniform_real distribution.
dist_uniform_real_quantile
Computes the quantile function (inverse CDF) of the uniform_real distribution.
dist_uniform_real_quantile_complement
Computes the complementary quantile function of the uniform_real distribution.
dist_uniform_real_range
Returns the range of the uniform_real distribution.
dist_uniform_real_sample
Generates random samples from the uniform_real distribution with specified parameters.
dist_uniform_real_skewness
Returns the skewness of the uniform_real distribution.
dist_uniform_real_stddev
Returns the standard deviation (σ) of the uniform_real distribution.
dist_uniform_real_support
Returns the support of the uniform_real distribution.
dist_uniform_real_variance
Returns the variance (σ²) of the uniform_real distribution.
dist_weibull_cdf
Computes the cumulative distribution function (CDF) of the weibull distribution.
dist_weibull_cdf_complement
Computes the complementary cumulative distribution function (1 - CDF) of the weibull distribution.
dist_weibull_chf
Computes the cumulative hazard function of the weibull distribution.
dist_weibull_hazard
Computes the hazard function of the weibull distribution.
dist_weibull_kurtosis
Returns the kurtosis of the weibull distribution.
dist_weibull_kurtosis_excess
Returns the excess kurtosis of the weibull distribution.
dist_weibull_log_cdf
Computes the natural logarithm of the cumulative distribution function (CDF) of the weibull distribution.
dist_weibull_log_cdf_complement
Computes the natural logarithm of the complementary cumulative distribution function (1 - CDF) of the weibull distribution.
dist_weibull_log_pdf
Computes the natural logarithm of the probability density function (log-PDF) of the weibull distribution.
dist_weibull_median
Returns the median (50th percentile) of the weibull distribution, which equals the mean.
dist_weibull_mode
Returns the mode (most likely value) of the weibull distribution, which equals the mean.
dist_weibull_pdf
Computes the probability density function (PDF) of the weibull distribution.
dist_weibull_quantile
Computes the quantile function (inverse CDF) of the weibull distribution.
dist_weibull_quantile_complement
Computes the complementary quantile function of the weibull distribution.
dist_weibull_range
Returns the range of the weibull distribution.
dist_weibull_sample
Generates random samples from the weibull distribution with specified parameters.
dist_weibull_skewness
Returns the skewness of the weibull distribution.
dist_weibull_support
Returns the support of the weibull distribution.
dist_weibull_variance
Returns the variance (σ²) of the weibull distribution.
Practical Examples
Cookbook
Real-world recipes and patterns for common use cases.
Recipes are organized by distribution family. Every distribution exposes the same operations — sample, pdf / log_pdf, cdf / log_cdf / cdf_complement, quantile, plus property functions (mean, variance, skewness, …). Once you’ve used one family, the rest follow the same shape.
Normal — the workhorse
Z-scores, confidence intervals, two-sided p-values, VaR. The Normal distribution is parameterized by mean and standard deviation:
-- Quantiles, density, CDF
SELECT dist_normal_quantile(0.0, 1.0, 0.95) AS p95; -- 1.6449
SELECT dist_normal_pdf(0.0, 1.0, 0.5) AS density;
SELECT dist_normal_cdf(0.0, 1.0, 1.96) AS prob; -- ~0.975
-- Z-score using property functions inline
SELECT
height_cm,
(height_cm - dist_normal_mean(100, 15))
/ dist_normal_stddev(100, 15) AS height_zscore
FROM observations;
-- Two-sided p-value from a Z-statistic
WITH z AS (SELECT 2.31 AS stat)
SELECT 2 * dist_normal_cdf_complement(0.0, 1.0, ABS(stat)) AS p_value
FROM z;See dist_normal_pdf, dist_normal_cdf, dist_normal_quantile, dist_normal_sample.
Discrete counts — Binomial and Poisson
Binomial for fixed-trial successes; Poisson for events at a rate:
-- Binomial: P(X = 7), P(X ≤ 5), random k
SELECT dist_binomial_pdf(10, 0.3, 7) AS prob_exactly_7;
SELECT dist_binomial_cdf(10, 0.3, 5) AS prob_at_most_5;
SELECT dist_binomial_sample(10, 0.3) AS k;
-- Poisson: probability of seeing >= 5 arrivals when λ=2.5
SELECT dist_poisson_cdf_complement(2.5, 4) AS prob_5_or_more;
-- Poisson sample for synthetic event-count columns
SELECT i, dist_poisson_sample(2.5) AS arrivals
FROM range(10) t(i);See dist_binomial_pdf, dist_binomial_cdf, dist_poisson_cdf, dist_poisson_sample.
Time-to-event — Exponential, Weibull, Gamma
Exponential for memoryless waiting times; Weibull for time-to-failure with non-constant hazard; Gamma for waiting times across multiple events:
-- Survival probability past t=2 for Exponential(rate=0.5)
SELECT dist_exponential_cdf_complement(0.5, 2.0) AS surviving;
-- Hazard rate at t=10 for Weibull(shape=1.5, scale=8)
SELECT dist_weibull_hazard(1.5, 8.0, 10.0) AS instantaneous_hazard;
-- Sample 1,000 service times from Gamma(2, 3)
SELECT dist_gamma_sample(2.0, 3.0) AS service_secs
FROM range(1000);See dist_exponential_cdf_complement, dist_weibull_hazard, dist_gamma_sample.
Bayesian conjugates — Beta and Gamma priors
Closed-form conjugate posteriors are one quantile call away. After observing 17 successes in 25 trials with a Beta(2, 2) prior, the posterior is Beta(19, 10):
-- Posterior mean and 95% credible interval
SELECT
dist_beta_mean(19, 10) AS posterior_mean,
dist_beta_quantile(19, 10, 0.025) AS ci_lo,
dist_beta_quantile(19, 10, 0.975) AS ci_hi;See dist_beta_mean, dist_beta_quantile.
Hypothesis testing — t, chi-squared, F
-- Two-sided t-test p-value, df=29
WITH t AS (SELECT 2.045 AS stat, 29 AS df)
SELECT 2 * dist_students_t_cdf_complement(df, ABS(stat)) AS p_value
FROM t;
-- Chi-squared goodness-of-fit p-value, df=4
WITH chi AS (SELECT 9.488 AS stat)
SELECT dist_chi_squared_cdf_complement(4, stat) AS p_value
FROM chi;
-- F-test (ANOVA) p-value, df1=3 df2=20
WITH f AS (SELECT 4.94 AS stat)
SELECT dist_fisher_f_cdf_complement(3, 20, stat) AS p_value
FROM f;See dist_students_t_cdf_complement, dist_chi_squared_cdf_complement, dist_fisher_f_cdf_complement.
Heavy-tailed families — Pareto, Cauchy, Log-normal
Wealth, file sizes, internet traffic, financial returns. The tail dominates; reach for these when the Normal underestimates extremes:
-- Probability a file exceeds 1 GB under Pareto(scale=10MB, shape=1.5)
SELECT dist_pareto_cdf_complement(10.0, 1.5, 1024.0) AS prob_over_1gb;
-- Median income under Log-normal(meanlog=10.5, sdlog=0.7)
SELECT dist_lognormal_median(10.5, 0.7) AS median_income;See dist_pareto_cdf_complement, dist_lognormal_median.
Synthetic data — combine families per column
Sampling functions return one draw per call, so combine them with range() over any source row count:
CREATE TABLE synthetic_users AS
SELECT
i AS user_id,
dist_normal_sample(170, 8) AS height_cm,
dist_lognormal_sample(10.5, 0.7) AS annual_income,
dist_poisson_sample(3.5) AS sessions_last_week,
dist_exponential_sample(0.1) AS avg_session_minutes,
dist_bernoulli_sample(0.12) AS converted
FROM range(100000) t(i);One query, 100k rows of realistic-looking fixture data — heights from a Normal, incomes from a Log-normal, session counts from a Poisson, conversions from a Bernoulli.
Monte Carlo — the canonical example
-- Estimate π by sampling uniform points in [-1,1]² and counting hits inside the unit circle
WITH samples AS (
SELECT dist_uniform_real_sample(-1.0, 1.0) AS x,
dist_uniform_real_sample(-1.0, 1.0) AS y
FROM range(1000000)
)
SELECT 4.0 * COUNT(*) FILTER (WHERE x*x + y*y <= 1) / COUNT(*) AS pi_estimate
FROM samples;The same pattern — generate, filter, aggregate — handles VaR, payoff integration, queueing simulation, A/B-test power calculations.
Value at Risk
-- 1-day 99% VaR assuming Normal returns
WITH params AS (SELECT 0.0001 AS mu, 0.02 AS sigma, 1000000 AS portfolio)
SELECT portfolio * (-dist_normal_quantile(mu, sigma, 0.01)) AS var_99
FROM params;For heavy-tailed VaR, swap dist_normal_quantile for dist_students_t_quantile with a low df.
Log-scale variants for likelihoods
Every PDF / CDF / CDF-complement has a _log_* variant. Use them inside likelihood computations and to avoid underflow on tiny tails:
-- Log-likelihood of one observation under N(0,1)
SELECT dist_normal_log_pdf(0.0, 1.0, 4.5) AS log_density;
-- Log-survival for very small tail probabilities
SELECT dist_normal_log_cdf_complement(0.0, 1.0, 6.0) AS log_tail;See dist_normal_log_pdf, dist_normal_log_cdf_complement.
Reproducibility
_sample functions don’t take a seed parameter. For exactly-reproducible runs, materialize the draws once into a table and treat that table as the seed:
CREATE TABLE rng_pool AS
SELECT i, dist_normal_sample(0, 1) AS z
FROM range(1000000) t(i);
-- Subsequent analyses read from rng_pool — fully reproduciblePlatform Support
Compatibility
Extension availability may vary by platform and DuckDB version. Check below to ensure this extension supports your environment before installation.
Quick Facts
Platforms
Supported platform architectures
Compiled binary sizes
| Platform | Architecture | Size |
|---|---|---|
| Linux | aarch64 | 3.85 MB |
| Linux | x86_64 | 4.38 MB |
| Linux (musl) | x86_64 | 3.69 MB |
| macOS | Apple Silicon | 3.15 MB |
| macOS | Intel | 3.71 MB |
| Windows | x86_64 | 8.59 MB |
| WASM | eh | 248.1 KB |
| WASM | mvp | 235.9 KB |
| WASM | threads | 235.8 KB |
Gzipped download size from the DuckDB community-extensions registry.
Monte Carlo, in SQL
Install Stochastic to compute PDFs, CDFs, quantiles, and random samples from 22 named probability distributions inside DuckDB — Monte Carlo and synthetic data without leaving the warehouse.