Question

Probability: Suppose that X1,X2...,Xn are i.i.d random variables, each with mean E(X) and Variance Var(x) . Let N be a random variable independent of the Xi's , N=0,1,2,3... . Let Sn= X1+X2+X3.....+Xn

Show that Var(Sn) =E(N)Var(X) +(E(x))^2 Var(N)

Answer 1

If I condition On N=n for example I get :

\[ Var(S_{n}| N=n)=Var(X_1)+Var(X_2)+..Var(X_n)\]
\[Var(S_n|N)=NVar(X), \] if I take the expectation I can get \[ E(N)Var(X)\] but I am still missing the second term, I must be doing something wrong

Answer 2

Here's my progress:
\[[E(S_n)]^2 = [E(N)E(X)]^2 = E^2(N)E^2(X)\]\[E(S_n^2) = E(N^2)E(X^2) = (Var(N)+E^2(N))(Var(X) + E^2(X))\]\[= Var(N)Var(X) + E^2(N)Var(X) + E^2(X)Var(N) + E^2(N)E^2(X)\]
\[Var(S_n) = E(S_n^2) - [E(S_n)]^2 = E(N^2)E(X^2) - E^2(N)E^2(X)\]\[= Var(N)Var(X) + E^2(N)Var(X) + E^2(X)Var(N)\]\[= (Var(N) + E^2(N))Var(X) + E^2(X)Var(N)\]\[= E(N^2)Var(X) + E^2(X)Var(N)\]Hmm...