Multinomial Distribution

The Multinomial distribution is a generalization of the Binomial. Whereas the Binomial distribution counts the successes in a fixed number of trails that can only be categorized as success or failure, the Multinomial distribution keeps track of trials whose outcomes can fall into multiple categories.

Let each of $n$ objects is independently placed into one of $k$ categories. An object is placed into category $j$ with probability $p_j$, where the $p_j$ are non-negative and ${\sum }_{j=1}^k{p}_j$. Let $X_i$ be the number of objects in category $i$, so that $X_1+\cdots +X_k=n$. Then $\text{X}=(X_1,\dots ,X_k)$ is said to have the Multinomial distribution with parameters $n$ and $\mathbf{p}=(p_1,\dots ,p_k)$. We write this as $\text{X}\sim {\mathrm{Mult}}_k(n,\mathbf{p})$

Then the joint PMF of $\text{X}$ is

$$P(X_1=n_1,\dots ,X_k=n_k)=\frac{n!}{n_1!n_2!\cdots n_k!}\cdot p_1^{n_1}{p}_2^{n_2}\cdots p_k^{n_k}$$

where $n_1+n_2+\cdots +n_k=n$

Property

• Mean: $\mathrm{E}[X_i]=n{p}_i$
• Variance: $\mathrm{Var}(X_i)=n{p}_i(1-p_i)$
• Covariance: $\mathrm{Cov}(X_i,X_j)=-n{p}_i{p}_j(i\ne j)$