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

transition

Required list input.

initial State

Required integer input.

count

Required integer input.

seed

Required string input.

Sample a finite-state Markov chain from a caller-supplied row-stochastic transition matrix and initial state. This samples the chain, not an arbitrary target 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]

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.

• Cauchy Distribution Sampler

  Sample a Cauchy distribution by inverse CDF from a seed. Cauchy samples have no finite mean or variance.

• Slice Sampler for Standard Normal

  Sample the standard normal density with univariate stepping-out and shrinkage slice sampling. Width controls exploration, not the target distribution.

• Bivariate Normal Gibbs Sampler

  Alternate conditional Gaussian draws for a zero-mean, unit-variance bivariate normal with caller-supplied correlation. Samples are autocorrelated.