The Multinomial distribution

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):

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

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

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

Arguments

Details

See also

Examples