Expectation of the product of two dependent binomial distributions.

Question

Let $X_i$ (i ∈ N) be independent and identically distributed all following a Bin(10, p) distribution for some value p ∈ [0, 1]. Define $Y_n$ := $$\sum_i^m X_i$$

Compute $t_n,m$ = Cov($Y_n$,$Y_m$) for n,m ∈ N where n <= m.

I apologise for my formatting; I hope the question is clear.

In calculating the covariance, I have easily been able to find E($Y_n$) and E($Y_m$), the issue is in calculating E($Y_nY_m$)

Any suggestions would be greatly appreciated.

Answer 1

We want the expectation of $$(X_1+X_2+\cdots +X_n)(X_1+X_2+\cdots +X_m),$$ where $n\le m$. So we want the expectation of $$(X_1+X_2+\cdots+X_n)^2 +(X_1+X_2+\cdots+X_n)(X_{n+1}+\cdots+X_m).$$ The second part is easy, by independence.

As for $E((X_1+X_2+\cdots+X_n)^2)$, the easiest way is to note that this is equal to the variance of $X_1+X_2+\cdots+X_n$ plus the square of $E(X_1+X_2+\cdots+X_n)$.