Question: Suppose $X1 \sim Binomial(n,p)$ and $X2 \sim Binomial(X1,p)$. Find the variance of X1 + X2.
This is my approach:
$Var(X1 + X2) = Var(X1) + Var(X2) + Cov(X1,X2)$
$Var(X1) = np(1-p)$
$Var(X2) = E[Var(X2|X1)]+ Var(E[X2|X1]) = E[p(1-p)X1] + Var(pX1) = np^{2}(1-p^{2})$
$Cov(X1,X2) = E(X1X2) - E(X1)E(X2) = E(X1X2) - n^{2}p^{3}$
At this point I have no idea how to calculate $E(X1X2)$ and I am hoping that I got the other steps correctly. Anyone can help me with this?
Firstly the formula for $\text{Var}(X_1+X_2)$ is given by $$\text{Var}(X_1+X_2)=\text{Var}(X_1)+\text{Var}(X_2)+2\text{Cov}(X_1,X_2)$$ We can find $\mathbb E(X_1X_2)$ by conditioning: $$\mathbb E(X_1X_2|X_1)=X_1\mathbb E(X_2|X_1)=X_1\cdot X_1p=X_1^2p$$ By the law of total expectation, $$\mathbb E(X_1X_2)=\mathbb E(\mathbb E(X_1X_2|X_1))=\mathbb E(X_1^2p)=p\mathbb E(X_1^2)$$ We find the second moment of $X_1$: $$\mathbb E(X_1^2)=\text{Var}(X_1)+\mathbb E(X_1)^2$$ and the result follows.