dist_multinomial: The Multinomial distribution

In distributional: Vectorised Probability Distributions

The Multinomial distribution

Description

The multinomial distribution is a generalization of the binomial distribution to multiple categories. It is perhaps easiest to think that we first extend a dist_bernoulli() distribution to include more than two categories, resulting in a dist_categorical() distribution. We then extend repeat the Categorical experiment several (n) times.

Usage

dist_multinomial(size, prob)

Arguments

size	The number of draws from the Categorical distribution.
prob	The probability of an event occurring from each draw.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_multinomial.html

In the following, let X = (X_1, ..., X_k) be a Multinomial random variable with success probability prob = p. Note that p is vector with k elements that sum to one. Assume that we repeat the Categorical experiment size = n times.

Support: Each X_i is in \{0, 1, 2, ..., n\}.

Mean: The mean of X_i is n p_i.

Variance: The variance of X_i is n p_i (1 - p_i). For i \neq j, the covariance of X_i and X_j is -n p_i p_j.

Probability mass function (p.m.f):

P(X_1 = x_1, ..., X_k = x_k) = \frac{n!}{x_1! x_2! \cdots x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot \ldots \cdot p_k^{x_k}

where \sum_{i=1}^k x_i = n and \sum_{i=1}^k p_i = 1.

Cumulative distribution function (c.d.f):

P(X_1 \le q_1, ..., X_k \le q_k) = \sum_{\substack{x_1, \ldots, x_k \ge 0 \\ x_i \le q_i \text{ for all } i \\ \sum_{i=1}^k x_i = n}} \frac{n!}{x_1! x_2! \cdots x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot \ldots \cdot p_k^{x_k}

The c.d.f. is computed as a finite sum of the p.m.f. over all integer vectors in the support that satisfy the componentwise inequalities.

Moment generating function (m.g.f):

E(e^{t'X}) = \left(\sum_{i=1}^k p_i e^{t_i}\right)^n

where t = (t_1, ..., t_k) is a vector of the same dimension as X.

Skewness: The skewness of X_i is

\frac{1 - 2p_i}{\sqrt{n p_i (1 - p_i)}}

Excess Kurtosis: The excess kurtosis of X_i is

\frac{1 - 6p_i(1 - p_i)}{n p_i (1 - p_i)}

See Also

stats::dmultinom(), stats::rmultinom()

Examples

dist <- dist_multinomial(size = c(4, 3), prob = list(c(0.3, 0.5, 0.2), c(0.1, 0.5, 0.4)))

dist
mean(dist)
variance(dist)

generate(dist, 10)

density(dist, list(d = rbind(cbind(1,2,1), cbind(0,2,1))))
density(dist, list(d = rbind(cbind(1,2,1), cbind(0,2,1))), log = TRUE)

cdf(dist, cbind(1,2,1))

distributional documentation built on June 27, 2026, 5:06 p.m.