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.
Best Answer
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 $X_i$.