Use this probability or sampling tool to compute a stated quantity or generate reproducible draws from a precisely parameterized distribution or stochastic model.

Gamma shape

Required number input.

Gamma scale

Required number input.

Sample count

Required integer input.

Seed

Required string input.

Generate reproducible overdispersed counts by mixing a Poisson rate with a gamma distribution. The tool evaluates the supplied inputs together and returns the named outputs below; it does not infer omitted operating conditions or change the units shown.^[1]

Mixture samples

The resulting mixture samples returned as a list.

• Results depend on parameterization, support, random source, seed, convergence, burn-in, autocorrelation, proposal quality, normalization, numerical tails, and whether model assumptions match the process. Samples are not exact evidence of convergence.
• Use finite inputs in the displayed units and preserve more precision than the final presentation requires. Independently verify safety-critical, financial, compliance, or production decisions.

Check analytic moments or known test cases, use independent chains and diagnostics for Monte Carlo methods, report seed and effective sample size, and inspect tail behavior.

• Negative Binomial–Poisson Mixture Sampler

  Draw a gamma-distributed Poisson rate and then an exact Poisson count; marginal counts follow the negative-binomial gamma-Poisson mixture in the supported rate range.

• Beta Distribution Sampler

  Generate reproducible beta-distributed values from two gamma variates.

• Gamma Distribution Sampler

  Generate reproducible gamma-distributed samples from a positive shape and scale.