Expectation of the product of two dependent binomial random variable

Question

Suppose you have n balls and m numbered boxes. You place each ball randomly and independently into one of the boxes. Let X_i be the number of balls placed into box number i, so $X_1 + · · · + X_m = n$.

For $i\ne j$, find $E(X_iX_j)$

This is what I have done so far... Can someone pointed out where I have go wrong, or give some hints on how to go forward.

Answer 1

Let $Y=X_1+\cdots+X_m$. Then $E(Y^2)=n^2$. But also $$E(Y^2)=\sum_1^m E(X_i^2)+\sum_{i\ne j}E(X_iX_j).$$ Now all we need is $E(X_i^2)$.

The random variable $X_i$ has binomial distribution, $p=1/n$, and number of trials equal to $m$. This has mean $\frac{m}{n}$, and variance $m\cdot \frac{1}{n}\cdot\left(1-\frac{1}{n}\right)$. So now we can calculate $E(X_i^2)$, and finish.