Determine the characteristic function of $X=\sum_{n=1}^{N} Y_{n}$, with $N \sim P(\lambda)$ and $Y_{i}$ i.i.d random variables

Question

I need to determine the characteristic function of $X$, $E(X)$ and $\operatorname{Var}(X)$, where $X=\sum_{n=1}^{N} Y_{n}$, with $N \sim P(\lambda)$ and $Y_{i}$ i.i.d random variables such as: $P(Y_{i}=1)=1/2$, $P(Y_{i}=2)=1 / 3$ and $P(Y_{i}=3)=1 / 6$.

This seems rather straightforward, but, for some reason, I can't seem to easily determine the characteristic function of $X$.

Answer 1

Suggestion:

Define $S_n=\sum^n_{k=1}Y_k$ so that $X=S_N$. Then

$\mathbb{E}[e^{itX}]=\mathbb{E}\big[\sum^\infty_{n=0}\mathbb{1}(N=n)e^{itS_N}\big]=\sum^\infty_{n=0}\mathbb{E}\big[\mathbb{1}(N=n)e^{itS_n}\big]$

Under the assumptions that $N$ is independent from $(Y_n:n\in\mathbb{N})$ and $(Y_n:n\in\mathbb{N})$ are i.i.d.

$$ \mathbb{E}\big[\mathbb{1}(N=n)e^{itS_n}\big]=\mathbb{P}[N=n]\Big(\mathbb{E}[e^{itY_1}]\Big)^n $$

I hope you can continue from this.