Solution
= Solution
By definition, the <probability generating function> is
$$
G_X(s)=\mathbb E(s^X)=\sum_{n=0}^{\infty}s^n\mathbb P(X=n).
$$
Substituting the <Poisson distribution> and evaluating the <exponential series> gives
$$
\boxed{G_X(s)=e^{-\lambda}\sum_{n=0}^{\infty}\frac{(\lambda s)^n}{n!}=e^{\lambda(s-1)}}.
$$
The series converges for every finite complex $s$; in particular it gives the <probability generating function> on $0\leq s\leq1$.