dist_multinomial {distributional} | R Documentation |
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.
dist_multinomial(size, prob)
size |
The number of draws from the Categorical distribution. |
prob |
The probability of an event occurring from each draw. |
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
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)