Suppose a coin is flipped 30 times. Let X = #heads in first 20 flips, Y = #heads in second 20 flips. I want to find Corr(X, Y).
I am only confused on how to find Cov( X, Y) = E[ XY] - E[ X]E[ Y], since I do not know how to calculate E[ XY]. My only idea towards solving is to make X = X_1 + X_2 and Y = Y_1 + Y_2 to denote the first, second, and third 10 coin flips. The variances are straightforward because X,Y~Binomial(20, .5).
We only need to find $E(XY)$, since we know that $E(X)=E(Y)=10$.
To do that, we use the decomposition idea mentioned in the question. Let $U$ be the number of heads in the first $10$ flips, $V$ the number of heads in the middle $10$ flips, and $W$ the number in the last $10$ flips.
Then $X=U+V$ and $Y=V+W$. We have $$E(XY)=E((U+V)(V+W))=E(UV)+E(UW)+E(V^2)+E(VW).$$ By independence, $E(UV)=E(UW)=E(VW)=25$.
To calculate $E(V^2)$, it is easiest to recall that $\text{Var}(V)=E(V^2)-(E(V))^2$. But $\text{Var}(V)=(10)(1/2)(1/2)$, so $E(V^2)=25+(10)(1/2)(1/2)$.