Hypergeometric Distribution
X ~ Hypergeometric(N, K, n)
Without-Replacement Sampling Family • Published July 20, 2026
The distribution for sampling without replacement: history from early urn problems through Fisher's 1934 exact test for contingency tables, the exact convergence to the already-published Binomial as the population grows large relative to the sample, the finite-population-correction variance formula, the classic Lincoln-Petersen mark-recapture maximum-likelihood estimator (with a verified likelihood tie at its boundary case), and simulation via direct without-replacement draws.
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About This Reference Sheet
The distribution for sampling without replacement: history from early urn problems through Fisher's 1934 exact test for contingency tables, the exact convergence to the already-published Binomial as the population grows large relative to the sample, the finite-population-correction variance formula, the classic Lincoln-Petersen mark-recapture maximum-likelihood estimator (with a verified likelihood tie at its boundary case), and simulation via direct without-replacement draws.
Support
x ∈ {max(0, n-(N-K)), ..., min(n, K)}
Parameters
N ∈ ℕ (population size), K ∈ {0,...,N} (successes in population), n ∈ {0,...,N} (draws)