Let $(X_i, P_i)$, with $_i$ being an integer be an independent random variable s.t. $X_i|P_i$ ~ $Bernoulli(P_i)$ and $P_i$ ~ $Beta(\alpha, \beta$.
Here's what I have so far: Var Y=Var(E(Y|P)+E(Var(Y|P) is what we know as our formula;
If we break down the two components on the right side we get: Var(E(Y|P))=$\alpha\beta/(\alpha+\beta)^2 (\alpha+\beta+1)$ and E(Var(Y|P)=$E(P_i(1-P_i))=\alpha/(\alpha+\beta) (1-\alpha/(\alpha+\beta))$.
Somehow I'm not getting the right result.
Is this correct and if not, could you please correct my understanding?