I am trying to proof the following: If $(X_n)_{n \in \mathbb{N}}$ is a sequence of discrete independent and identically distributed random variables and if N is a integer valued random variable which is independent of the sequence then $Var(X_1+...X_N)=E[N](Var(X_i))^2+Var(N)(E[X_i])^2$ holds.
My try was:
$Var(X_1+...X_N)=E((X_1+...+X_N)^2)-(E(X_1+...+X_N))^2$= $E[(X_1+...+X_N)^2]-(E[N]*E[X_i])^2$= $E[(N*X_i)^2]-(E[N]*E[X_i])^2$=$E[N^2]*E[(X_i)^2]-E[N]^2*E[X_i]^2$
thanks in advance