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

Multinomial(size, p)

Arguments

size

The number of trials. Must be an integer greater than or equal to one. When size = 1L, the Multinomial distribution reduces to the categorical distribution (also called the discrete uniform). Often called n in textbooks.

p

A vector of success probabilities for each trial. p can take on any positive value, and the vector is normalized internally.

Value

A Multinomial object.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail and much greater clarity.

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):

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

See also

Examples

set.seed(27) X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1)) X
#> Multinomial distribution (size = 5, p = 0.3) Multinomial distribution (size = 5, p = 0.4) Multinomial distribution (size = 5, p = 0.2) Multinomial distribution (size = 5, p = 0.1)
random(X, 10)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] #> [1,] 5 5 5 5 5 5 5 5 5 5
# pdf(X, 2) # log_pdf(X, 2)