MultinomialDistribution

$n$ draws over $k$ categories with probabilities $p_1,\ldots,p_k$ (summing to 1); a count vector on the $n$ -simplex.

MultinomialDistribution(n, {p1, ..., pk})n independent draws, each landing in category i with probability pi.

Typed signatures

Type	Package	Applies
`(real<0..>, list<real>) -> distribution`	`statistics`	always

Domain: Statistics

Details

$PDF(x) = \binom{n}{x_1,\ldots,x_k} \prod_i p_i^{x_i}$ when $\sum_i x_i = n$ , $0$ otherwise — the multinomial coefficient is compute-engine's own native Multinomial(x1,...,xk), which already computes $n!/(x_1!\cdots x_k!)$ for $n = \sum x_i$ ; the guard is what checks that sum against the DISTRIBUTION's own $n$ .
$Mean_i = n\,p_i$ ; $Variance_i = n\,p_i(1-p_i)$ — both length- $k$ lists (Wolfram's own Variance convention: componentwise, not the full covariance matrix).
Covariance (compute-engine native over two collections) is widened to also accept a single MultinomialDistribution, answering the full $k \times k$ covariance matrix: $Cov(X_i,X_j) = -n\,p_i p_j$ ( $i \ne j$ ), $Var(X_i) = n\,p_i(1-p_i)$ on the diagonal.
RandomVariate draws via sequential conditional binomials (category $i$ gets $Binomial(\text{remaining } n, p_i / \text{remaining } p)$ successes, the last category takes the exact remainder), seeded through the same SeedRandom stream every other distribution in this package uses.