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:


 * $$\Pi_X \left({s}\right) = \sum_{x \ge 0} p_X \left({x}\right) s^x$$

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

So:

$$ $$ $$ $$

Hence the result.