Probability Generating Functions = Fonctions Génératrices

Formula for the PGF:

Let $X$ be a random variable that takes values in $\N$. For $s \in [0, 1[$

Why bothering?

• $g_X(0) = P(X = 0)$

• $\frac{g_X^{(n)}}{n!}(0) = P(X = n)$

• $g_X(1) = \sum_{n \ge 0}^{} P(X = n) = 1$

• $g_X^{(k)}(1) = E[X(X-1)\ldots(X-k+1)]$

That is why it describes a distribution.