# Mean, Variance and Covariance of Multinomial Distribution

I'm working through the following problem:

Let $(X_1, \dots , X_k)$ be a random vector with multiomial distribution $\mathcal{M}(p_1, \dots , p_k, n)$. Determine $\mathbb{E}(X_i)$, $Var(X_i)$ and $Cov(X_i, X_j)$ with $i \neq j$ for each $1 \leq i, j \leq n$.

Given the probabilty function for the vector,

$$p(x_1, \dots , x_n) = \frac{n}{x_1! \cdot \dots \cdot x_n!}p_1^{x_1}\cdot \dots \cdot p_k^{x_k}$$

if $x_1 + \dots x_n = n$, and zero otherwise, I've tried rewriting this in such a way that I can recover the probability function for $X_i$ (is that even possible without asking for independence?), by summing on the other variables, and so I can at least calculate the mean, and then go from there. So far I've had no success. Is this the correct approach? Or is there a more elegant way to go about this?

P.D.: it may seem like I haven't tried enough. I'm fairly new to probability theory and for some reason I have a tough time gaining intuition in this subject. I've dedicated this problem some time, but I feel like I am conceptually stuck, and in need of a hint. Thanks in advance for your help.

The multinomial distribution corresponds to $n$ independent trials where each trial has result $i$ with probability $p_i$, and $X_i$ is the number of trials with result $i$. Let $Y_{ij}$ be $1$ if the result of trial $j$ is $i$, $0$ otherwise. Thus $X_i = \sum_{j} Y_{ij}$. It is easy to compute the means, variances and covariances of $Y_{ij}$ and use them to compute the means, variances and covariances of $X_i$.

• Thanks! I particularly appreciate this answer because it gives me more context and intuition. I'll try this out and get back to you, if you don't mind checking my work Commented Oct 22, 2017 at 17:56

Multinomial distribution function for $n$ random variables $\{x_{i}^{}\}$ is given by : $$P[\{x_{i}^{}\}]=n!\prod_{i=1}^{n}\frac{p_{i}^{x_{i}^{}}}{x_{i}^{}!},$$ with $0 \leq p_{i}^{} \leq 1$. Moment generating function for multinomial distribution is : $$\mathcal{Z}[\{\lambda_{i}^{}\}]=\mathop{\sum_{x_{1}^{}=1}^{n} \dots \sum_{x_{n}^{}=1}^{n}}_{\sum_{i}^{n}x_{i}=n}^{}n!\prod_{i=1}^{n}\frac{p_{i}^{x_{i}^{}}}{x_{i}^{}!}e^{i\lambda_{i}x_{i}^{}}_{},$$ which can be simplified using multinomial theorem as $$\mathcal{Z}[\{\lambda_{i}^{}\}]=\Big[\sum_{i=1}^{n} p_{i}^{} e^{i\lambda_{i}^{}}\Big]^{n}_{}.$$ All the moments of the random variables $\{x_{i}^{}\}$ can be obtianed as : $$\langle x_{a_1} \dots x_{a_m}\rangle = (-i)^{m}_{}\frac{\partial}{\partial \lambda_{a_1}}\dots \frac{\partial}{\partial \lambda_{a_m}} \mathcal{Z}[\{\lambda_{i}^{}\}] \Big|_{\{\lambda_{i}^{}\}=\{0\}}.$$ From which central moments can be obtained. For obtaining cumulants you have to take derivatives of $\log\mathcal{Z}[\{\lambda_{i}^{}\}]$.