Multinomial Distribution

def. Multinomial Distribution. For outcome space $\Omega$, $k$ mutually exclusive events each with probabilities $p_1,\dots ,p_k$ s.t. ${\sum }_{i=1}^k{p}_i=1$, let random variables $X_i$ denote the number of outcomes whose probability is $p_i$, when $n$ trials are done:

$$\begin{gathered} X_1,...,X_k\sim \text{Multinomial}(n,k,p_1,...,p_k) \\ P(X_1=x_1,...,X_k=x_k)=\frac{n!}{x_1!,...,x_k!}\cdot p_1^{x_1}\times \cdots \times p_k^{x_k} \end{gathered}$$

⇒ Multinomial is a generalization of the Binomial Distribution.