multinomial

Multinomial is a term used in mathematics to describe several related concepts that generalize the binomial case to more than two categories. It appears prominently in combinatorics and probability, with the multinomial theorem, multinomial coefficients, and the multinomial distribution as core ideas.

The multinomial coefficient generalizes the binomial coefficient. For nonnegative integers n1, n2, ..., nk with sum n, the multinomial coefficient is n! / (n1! n2! ... nk!). It counts the number of ways to distribute n distinct objects into k labeled categories with specified counts n1 through nk.

The multinomial theorem extends the binomial theorem to multiple variables. For variables x1, x2, ..., xk and a nonnegative integer n, (x1 + x2 + ... + xk)^n equals the sum over all nonnegative integer tuples (n1, ..., nk) with n1 + ... + nk = n of the multinomial coefficient times the product x1^n1 x2^n2 ... xk^nk. The multinomial coefficient appears as the coefficient of each monomial in the expansion.

In probability, the multinomial distribution describes the outcomes of n independent trials, each trial resulting in one of k categories with probabilities p1, ..., pk (sum to 1). If X1, ..., Xk denote the counts of each category, then (X1, ..., Xk) follows Multinomial(n, p1, ..., pk). The probability of a specific count vector is n!/(n1! ... nk!) ∏ pi^ni. The expectations satisfy E[Xi] = n pi, variances Var(Xi) = n pi (1 - pi), and covariances Cov(Xi, Xj) = -n pi pj for i ≠ j. When k = 2, the multinomial distribution reduces to the binomial distribution.

Applications span genetics, natural language processing, market research, and survey analysis. The multinomial distribution is related to the Dirichlet-multinomial model and is a fundamental tool for modeling counts across multiple categories.