Let $N, X_1, X_2, . . .$ be independent random variables such that $N ∈ Pois(1)$ and $X_k ∈ Pois(2)$ for all $k$. Set $Z = \sum_{k=1}^{N}X_k$ (and $Z = 0$ when $N = 0$). Compute $E(Z), Var(Z),$ and $P(Z = 0)$.
So, $X_k$ and $N$ are Poisson r.v. $f(X_k=k)=e^{-1}\frac{1}{k!}$; $f(N=k)=e^{-2}\frac{2}{k!}$.
Probability generating functions:
$g_{z}(t)=g_n(g_x(t))$, where $g_x(t)=e^{-1}\sum_{k=0}t^k/k!=e^{-1+t}$,$g_n(t)=2e^{-2}\sum_{k=0}t^k/k!=2e^{-2+t}$
So, $g_z(t)=2e^{-2+e^{-1+t}}$, now $E(Z)=\frac{dg_z(t)}{dt}$ at $t=1$; $\frac{dg_z(t)}{dt}=2e^{e^{t-1}+t-3}$ and at $t=1$, I recieve $E(Z)=2e^{1-2}$
the answer is $E(Z)=2$
Can you help me? What is wrong in the computations? Also $P(Z=k)=\frac{g_z^k(t)}{k!}$ is this right?