Expected value of a summation of random variable when $N$ is also a RV

Question

What would be the expected value of $S$ and its variance given $S$ such as a follow:

$$ S = \sum_{i=1}^{N} X_i, $$

and here $X_i$ and $N$ are both random variable.

Answer 1

Assuming $X$ and $N$ are independent, let $S_N=\sum_{i=1}^NX_i$, then $E(S_N|N=n)=\sum_{i=1}^nE(X_i)$ Let $P_n=prob(N=n)$, then $E(S)=\sum_{n=1}^\infty P_n\sum_{i=1}^nE(X_i)$