Multinomial Distribution
X ~ Multinomial(n, p)
Multi-Category Trials Family • Published July 20, 2026
The first multivariate sheet in this series -- the direct multi-category generalization of the already-published Binomial behind Pearson's chi-square test: an exact construction as independent Poisson counts conditioned on a fixed sum, the full covariance structure showing every pair of categories is negatively correlated, a verified (k-1)x(k-1) Fisher information matrix, a rare fully closed-form maximum-likelihood estimator (the sample proportion in each category), and simulation via the Poisson-conditioned-on-sum construction.
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About This Reference Sheet
The first multivariate sheet in this series -- the direct multi-category generalization of the already-published Binomial behind Pearson's chi-square test: an exact construction as independent Poisson counts conditioned on a fixed sum, the full covariance structure showing every pair of categories is negatively correlated, a verified (k-1)x(k-1) Fisher information matrix, a rare fully closed-form maximum-likelihood estimator (the sample proportion in each category), and simulation via the Poisson-conditioned-on-sum construction.
Support
(x1,...,xk) with xi ≥ 0 integers, x1+...+xk = n
Parameters
n ≥ 1 trials (integer), p = (p1,...,pk) with pi ≥ 0, Σpi = 1