We recommend reading this documentation on pkgdown which renders math nicely.
https://pkg.mitchelloharawild.com/distributional/reference/dist_multinomial.html
In the following, let \(X = (X_1, ..., X_k)\) be a Multinomial
random variable with success probability prob = \(p\). 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):
$$
P(X_1 = x_1, ..., X_k = x_k) = \frac{n!}{x_1! x_2! \cdots x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot \ldots \cdot p_k^{x_k}
$$
where \(\sum_{i=1}^k x_i = n\) and \(\sum_{i=1}^k p_i = 1\).
Cumulative distribution function (c.d.f):
$$
P(X_1 \le q_1, ..., X_k \le q_k) = \sum_{\substack{x_1, \ldots, x_k \ge 0 \\ x_i \le q_i \text{ for all } i \\ \sum_{i=1}^k x_i = n}} \frac{n!}{x_1! x_2! \cdots x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot \ldots \cdot p_k^{x_k}
$$
The c.d.f. is computed as a finite sum of the p.m.f. over all integer vectors
in the support that satisfy the componentwise inequalities.
Moment generating function (m.g.f):
$$
E(e^{t'X}) = \left(\sum_{i=1}^k p_i e^{t_i}\right)^n
$$
where \(t = (t_1, ..., t_k)\) is a vector of the same dimension as \(X\).
Skewness: The skewness of \(X_i\) is
$$
\frac{1 - 2p_i}{\sqrt{n p_i (1 - p_i)}}
$$
Excess Kurtosis: The excess kurtosis of \(X_i\) is
$$
\frac{1 - 6p_i(1 - p_i)}{n p_i (1 - p_i)}
$$
![[Stable]](figures/lifecycle-stable.svg?package=distributional&version=0.6.0)