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:
 * $\Pi_X \left({s}\right) = e^{-\lambda \left({1-s}\right)}$

Proof
From the definition of p.g.f:
 * $\displaystyle \Pi_X \left({s}\right) = \sum_{x \ge 0} p_X \left({x}\right) s^x$

From the definition of the Poisson distribution:
 * $\displaystyle \forall k \in \N, k \ge 0: p_X \left({k}\right) = \frac {e^{-\lambda} \lambda^k} {k!}$

So:

Hence the result.