Multinomial Distribution

Definition

$X\sim \mathrm{Mult}(n,p_1,p_2,\dots ,p_{k-1})$ where $n$ is the number of trials, $p_i$ is a probability of category $i$

Distribution that describes the probability of observing a specific combination of outcomes

Properties

PDF

$f(x_1,x_2,\dots ,x_{k-1})=n!{\prod }_{i=1}^k\frac{p_i^{x_i}}{x_i!}$ where $x_k=n-\sum _{i=1}^{k-1}{x}_i$, $p_k=1-\sum _{i=1}^{k-1}{p}_i$, and $0\le \sum _{i=1}^{k-1}{x}_i\le n$

MGF

$M(t_1,t_2,\dots ,t_{k-1})=({\sum }_{i=1}^{k-1}{p}_i{e}^{t_i}+p_k{)}^n$

Marginal PDF

Each one-variable marginal pdf is Binomial Distribution, each two-variables marginal pdf is Trinomial Distribution, and so on.