Probability Generating Function of the difference between 2 Poisson random variables

Question

Suppose we have 2 Poisson random variables, X and Y where X ~ Poisson(a) and Y ~ Poisson(b). What would be the probability generating function of X - Y then? I think the first step is to realize that this probability generating function = E[$z^{X-Y}$] = E[$z^X$]E[$z^{-Y}$]. However, I cannot proceed any further after this.

Answer 1

You have to assume that $X$ and $Y$ are independent. Otherwise you cannot say that $Ez^{(X-Y)}=Ez^{X}Ez^{-Y}$ and, in fact, you cannot compute the generating function of $X-Y$ at all.

$Ez^{X} = \sum\limits_{k=0}^{\infty} e^{-a} \frac {z^{n}a^{n}} {n!}=e^{-a}\sum\limits_{k=0}^{\infty} \frac {(az)^{n}} {n!}=e^{-a+az}$. Similarly compute $Ez^{-Y}$.