Multinomial distribution

• $N \in \mathbb{N}$,
  • given
• $K \in \mathbb{N}$,
  • given
• $p_{i} \in [0, 1]$,
  • given
  • probability
• \([0:K] := \{0, 1, \ldots, K\}\),
• \(X_{1}, \ldots, X_{N} \in [0:K]\),
  • r.v.

\[\mathcal{A} := \{ (x_{1}, \ldots, x_{N}) \in [0:K]^{N} \mid \sum_{i=1}^{N} x_{i} = K \} .\]

$\mathrm{Mult}(x_{1}, \ldots, x_{N}; p_{1}, \ldots, p_{N})$ is a p.d.f. of binomial distribution given $p$ and $N$ defined by

\[(x_{1}, \ldots, x_{N}) \in \mathcal{A}, \ \mathrm{Mult}(x_{1}, \ldots, x_{N}; p) := \frac{ K! }{ x_{1}! \cdots x_{N}! } p_{1}^{x_{1}} \dots p_{N}^{x_{N}} .\]

Reference