Suppose $X\sim\operatorname{Po}(\lambda)$ and $Y\sim\operatorname{Po}(\mu)$. Given $Z=X+Y$ , find the $\mathbb{E}[Z]$ and $\operatorname{Var}[Z]$ by first finding the generating function of $Z$.
I've already found the generating function of $Z$ to be $\exp(\lambda(t\exp(\mu(t-1))-1)$.
I'm just really lost in how to find the expectation and variance. I'm not sure if I should differentiate them. Any advice or guided steps would be great!
Edit (28/10/18) : I already got the answer with the help of Clement C and doing some more digging. $\mathbb{E}[Z]$ = $(\lambda)(1-\mu)$ and $\operatorname{Var}[Z]$= $\lambda(1+3\mu+\mu^2)$ . Thank you! (answers I have are correct with double checking across google. just needed to know the middle steps)
Recall that if the MGF is sufficiently differentiable at $0$, $$ \frac{d}{dt} \mathbb{E}[e^{tZ}]\Big|_0 = \mathbb{E}\left[ \frac{d}{dt} e^{tZ} \Big|_0 \right] = \mathbb{E}\left[ Z e^{tZ} \Big|_0 \right] = \mathbb{E}\left[ Z e^{0} \right] = \mathbb{E}\left[ Z \right] $$ while $$ \frac{d^2}{dt^2} \mathbb{E}[e^{tZ}]\Big|_0 = \mathbb{E}\left[ \frac{d^2}{dt^2} e^{tZ} \Big|_0 \right] = \mathbb{E}\left[ Z^2 e^{tZ} \Big|_0 \right] = \mathbb{E}\left[ Z^2 \right] $$ so that computing the first and second derivatives of the MGF you found at $0$ will give you the first and second moments of the r.v., and from there you have the expectation (first moment) and can easily obtain the variance.
(Switching differentiation and expectation requires justification, but I am assuming you can take it from granted from your class or textbook.)