I came across an interesting problem in Jacod's probability book. But have no idea how to approach it. Should I approach it using induction? Any ideas?
Let $X_1, X_2, \cdots$ be an infinite sequence of iid sequence of integrable random variables and let $N$ be a positive, integer-valued integrable random variable which is independent from the sequence. Define $S_n = \sum_{k=1}^{n} X_k$ and assume that $S_0 = 0$.
(a) Show that $E[S_N] = E[N]E[X_1]$.
(b) Show that the characteristic function of $S_N$ is given by $E[\phi_{X_{1}}(t)^N]$, where $\phi_{X_{1}}$ is the characteristic function of $X_1$.