Solution
= Solution
For a nonnegative integer-valued <random variable>, its <probability generating function> is $G_X(s)=\mathbb E[s^X]=\sum_{k\geq0}\mathbb P(X=k)s^k$, at least for $|s|\leq1$. Insert the <Poisson distribution> masses and sum the <exponential series>:
$$
G_X(s)=e^{-\mu}\sum_{k=0}^\infty\frac{(\mu s)^k}{k!}=\boxed{e^{\mu(s-1)}}.
$$