Multivariate Hypergeometric Distribution
X ~ MVHypergeom(N, K, n)
Multi-Category Trials Family • Published July 22, 2026
The missing without-replacement counterpart to the already-published Multinomial, and the direct multi-category generalization of the already-published Hypergeometric: draws without replacement from a population split into k labeled categories. Confirmed numerically to converge to the Multinomial as the population grows large relative to the sample, with a closed-form method-of-moments estimator for the category sizes standing in for a discrete, no-closed-form maximum-likelihood search.
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About This Reference Sheet
The missing without-replacement counterpart to the already-published Multinomial, and the direct multi-category generalization of the already-published Hypergeometric: draws without replacement from a population split into k labeled categories. Confirmed numerically to converge to the Multinomial as the population grows large relative to the sample, with a closed-form method-of-moments estimator for the category sizes standing in for a discrete, no-closed-form maximum-likelihood search.
Support
(x1,...,xk) integers, 0 ≤ xi ≤ Ki, x1+...+xk = n
Parameters
N population size, K=(K1,...,Kk) with ΣKi=N, n ≤ N draws