Q:

Variance of the compound sum

Trying to solve a Variance evaluation problem:

Now I'm not sure how to evaluate those two terms on the right hand side of the last equality...
Would appreciate any help.

A:

So, $V(\mu N)=\mu ^2 V(N)$ and $Var (S|N=n) = Var (\sum_{i=1}^N X_i|N=n)\\=Var (\sum_{i=1}^n X_i|N=n)\\=Var (\sum_{i=1}^n X_i)\\=\sum_{i=1}^n Var (X_i)\\=n Var (X_i)\\ =n\sigma^2$. 
Last lines are due to i.i.d nature of $X_i$'s.
So, $E(Var (S|N))=E(N\sigma^2)=\sigma^2E(N)$
This is known as Wald equation inliterature.

