Solution
= Solution
For a nonnegative integer-valued <random variable>, the <probability generating function> is
$$
G_X(s)=\mathbb E[s^X]=\sum_{n=0}^{\infty}\mathbb P(X=n)s^n,
$$
initially for $|s|\le1$, with $0^0=1$. For the <Poisson distribution>, insert its <probability mass function> and sum the exponential series:
$$
G_X(s)=e^{-\lambda}\sum_{n=0}^{\infty}\frac{(\lambda s)^n}{n!}.
$$
Thus
$$
\boxed{G_X(s)=e^{\lambda(s-1)}.}
$$