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 https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

In the following, let X = (X_1, ..., X_k) be a Multinomial random variable with success probability p = 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! ... x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot ... \cdot p_k^{x_k}

Cumulative distribution function (c.d.f):

Omitted for multivariate random variables for the time being.

Moment generating function (m.g.f):

E(e^{tX}) = \left(\sum_{i=1}^k p_i e^{t_i}\right)^n

See Also

stats::Multinomial

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)

# TODO: Needs fixing to support multiple inputs
# density(dist, 2)
# density(dist, 2, log = TRUE)

distributional documentation built on March 31, 2023, 7:12 p.m.