Probability Generating Function of Poisson Distribution

Theorem
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$.

Then the p.g.f. of $X$ is:
 * $\map {\Pi_X} s = e^{-\lambda \paren {1 - s} }$

Proof
From the definition of p.g.f:
 * $\ds \map {\Pi_X} s = \sum_{x \mathop \ge 0} \map {p_X} x s^x$

From the definition of the Poisson distribution:
 * $\ds \forall k \in \N, k \ge 0: \map {p_X} k = \frac {e^{-\lambda} \lambda^k} {k!}$

So:

Hence the result.