distributional (version 0.5.0)

dist_multinomial: 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)