dist_multinomial
The Multinomial distribution
Description
![[Maturing]](../help/figures/lifecycle-maturing.svg)
Usage
dist_multinomial(size, prob)
Arguments
size |
integer, say N, specifying the total number
of objects that are put into K boxes in the typical multinomial
experiment. For |
prob |
numeric non-negative vector of length K, specifying the probability for the K classes; is internally normalized to sum 1. Infinite and missing values are not allowed. |
Details
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 categorical distribution.
We then extend repeat the Categorical experiment several (n)
times.
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) = n! / (x_1! x_2! ... x_k!) p_1^x_1 p_2^x_2 ... 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)) = (p_1 e^t_1 + p_2 e^t_2 + ... + p_k e^t_k)^n
See Also
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)