Deriving the distribution of poisson random variables. Let $X_1, \dots, X_n$ be independent and identically distributed Poisson(µ) random variables.
Derive the distribution of $W= \sum X_i$.
I'm not sure how to answer this, should I use the moment generating function? 

Yes, we can use the moment generating functions to prove this. We have that
$$
M_W(t)=M_{X_1}(t)M_{X_2}(t)\ldots M_{X_n}(t)=[\exp(\mu(e^t-1))]^n=\exp(n\mu(e^t-1))
$$
using the independence and identical distributions. This means that the sum is a Poisson random variable with the parameter equal to $n\mu$.
