Stochastic

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;

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: 1.500000

-- 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;

-- 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);

-- 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);

-- 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;

-- 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;

-- 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;

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);

-- 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;

-- 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;

-- 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;

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 reproducible