Multinomial: Create a 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 Bernoulli() distribution to include more than two categories, resulting in a Categorical() distribution. We then extend repeat the Categorical experiment several ($n$) times.

Usage

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.

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):

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

Examples

Run this code

# NOT RUN {
set.seed(27)

X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X

random(X, 10)

# pdf(X, 2)
# log_pdf(X, 2)

# }

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