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)

## 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):

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

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
#> <distribution[2]>
#> [1] Multinomial(4)[3] Multinomial(3)[3]
mean(dist)
#>      [,1] [,2] [,3]
#> [1,]  1.2  2.0  0.8
#> [2,]  0.3  1.5  1.2
variance(dist)
#>      [,1] [,2] [,3]
#> [1,] 0.84 1.00 0.64
#> [2,] 0.27 0.75 0.72

generate(dist, 10)
#> [[1]]
#>       [,1] [,2] [,3]
#>  [1,]    1    3    0
#>  [2,]    2    1    1
#>  [3,]    1    3    0
#>  [4,]    1    3    0
#>  [5,]    0    4    0
#>  [6,]    1    2    1
#>  [7,]    1    3    0
#>  [8,]    4    0    0
#>  [9,]    1    2    1
#> [10,]    2    2    0
#>
#> [[2]]
#>       [,1] [,2] [,3]
#>  [1,]    1    1    1
#>  [2,]    1    2    0
#>  [3,]    0    2    1
#>  [4,]    0    1    2
#>  [5,]    0    3    0
#>  [6,]    1    0    2
#>  [7,]    0    2    1
#>  [8,]    0    1    2
#>  [9,]    0    2    1
#> [10,]    0    1    2
#>

# TODO: Needs fixing to support multiple inputs
# density(dist, 2)
# density(dist, 2, log = TRUE)