Negative Hypergeometric Distribution
X ~ NegHyper(N, K, r)
Without-Replacement Sampling Family • Published July 22, 2026
The missing corner of the 2×2 sampling-design table: draws without replacement until a fixed number of failures, filling in exactly where the already-published Hypergeometric (fixed sample size, without replacement) and Negative Binomial (fixed failures, with replacement) leave off, with a discrete-parameter maximum-likelihood estimator of the population's success count derived via the same ratio-of-likelihoods technique as the Hypergeometric sheet's own estimator, and simulation via the distribution's own without-replacement draw.
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About This Reference Sheet
The missing corner of the 2×2 sampling-design table: draws without replacement until a fixed number of failures, filling in exactly where the already-published Hypergeometric (fixed sample size, without replacement) and Negative Binomial (fixed failures, with replacement) leave off, with a discrete-parameter maximum-likelihood estimator of the population's success count derived via the same ratio-of-likelihoods technique as the Hypergeometric sheet's own estimator, and simulation via the distribution's own without-replacement draw.
Support
x ∈ {0, 1, ..., K}
Parameters
N ∈ ℕ (population size), K ∈ {0,...,N} (successes in population), r ∈ {1,...,N-K} (failures to stop at)