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

shape

Required number input.

success Probability

Required number input.

count

Required integer input.

seed

Required string input.

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. 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]

rates

The resulting rates returned as a list.

samples

The resulting 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.

• Poisson-Gamma Mixture Sampler

  Generate reproducible overdispersed counts by mixing a Poisson rate with a gamma distribution.

• Negative Binomial Distribution Sampler

  Draw the number of failures before a specified integer number of successes in independent Bernoulli trials.

• Poisson Distribution Sampler

  Generate reproducible Poisson-distributed event-count samples.