Multinomial distribution

• $N\in \mathbb{N}$,
  • given
• $K\in \mathbb{N}$,
  • given
• $p_i\in [0,1]$,
  • given
  • probability
• $[0:K]:=\{0,1,\dots ,K\}$,
• $X_1,\dots ,X_N\in [0:K]$,
  • r.v.

$$\mathcal{A}:=\{(x_1,\dots ,x_N)\in [0:K{]}^N\mid \sum _{i=1}^N{x}_i=K\}.$$

$\mathrm{Mult}(x_1,\dots ,x_N;p_1,\dots ,p_N)$ is a p.d.f. of binomial distribution given $p$ and $N$ defined by

$$(x_1,\dots ,x_N)\in \mathcal{A},\mathrm{Mult}(x_1,\dots ,x_N;p):=\frac{K!}{x_1!\cdots x_N!}{p}_1^{x_1}\dots p_N^{x_N}.$$

Reference