sum of independent random variables where $N$ is a random variable

Question

I want to show $E[S_N]=E[N]E[X_j]$ where:

$X_1,X_2,\ldots$ is a sequence of independent random variables, and $N$ is a random variable independent of the sequence. $S_n=\sum_{i=1}^n X_i$, $S_N=X_1+X_2+\ldots+X_N$,

So far, I have $$E[S_N]=E\left[\sum_{n=0}^{\infty} S_n1_{\{N=n\}}\right]=\sum_{n=0}^{\infty}E[S_n1_{\{N=n\}}]$$ by monotone convergence theorem,

$$=\sum_{n=0}^{\infty}E[S_n]P(N=n)$$ by the fact that $S_n$ and $1_{N=n}$ is independent since $N$ is independent from the sequence so $N$ is independent from their sum.

$$=\sum_{n=0}^{\infty}E\left[\sum_{j=1}^n X_j\right] P(N=n)$$ $$=\sum_{n=0}^{\infty}\left[\sum_{j=1}^n E[X_j]\right] P(N=n)$$

since $X_j$ are independent. I'm not sure if I can switch the summations here since the one inside goes to $n$.

Answer 1

The proposition only makes sense if all the X_j have the same expectation. Then the sum over j is just a multiplication by n. This completes the proof.