Multinomial Distribution

Definition

$\mathbf{X}\sim \text{Multinomial}(n;p_1,p_2,\dots ,p_k)$

Extension of binomial distribution to $k$ categories.

$x_i$ is the number of success in the $i$-th variable.

• pmf: $\frac{n!}{x_1!\cdots x_k!}{p}_1^{x_1}\cdots p_k^{x_k}$
  • $x_k=n-(x_1+\cdots +x_{k-1})$
  • $p_k=1-(p_1+\cdots +p_{k-1})$
• Mean: $E(X_i)=n{p}_i$
• Variance: $\text{Var}(X_i)=n{p}_i(1-p_i)$
• Covariance: $\text{Cov}(X_i,X_j)=-n{p}_i{p}_j$