Properties of Probability Generating Function

Theorem

Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$.

Then $\Pi_X \left({s}\right)$ has the following properties:

Probability Generating Function defines Probability Distribution

Let $X$ and $Y$ be discrete random variables whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$.

Let the probability generating functions of $X$ and $Y$ be $\map {\Pi_X} s$ and $\map {\Pi_Y} s$ respectively.

Then:

$\forall s \in \closedint {-1} 1: \map {\Pi_X} s = \map {\Pi_Y} s$

if and only if:

$\forall k \in \N: \Pr \left({X = k}\right) = \map \Pr {Y = k}$

That is, discrete random variables which are integer-valued have the same PGFs if and only if they have the same PMF.

PGF of 0

$\map {\Pi_X} 0 = \map {p_X} 0$

PGF of 1

$\map {\Pi_X} 1 = 1$