Probability Generating Function defines Probability Distribution

Theorem

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:

$\forall s \in \closedint {-1} 1: \map {\Pi_X} s = \map {\Pi_Y} s$
$\forall k \in \N: \Pr \left({X = k}\right) = \map \Pr {Y = k}$

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

Proof

By the definition of PGF, it follows that if:

$\forall k \in \N: \map \Pr {X = k} = \map \Pr {Y = k}$

then:

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

Suppose that $\map {\Pi_X} s = \map {\Pi_Y} s$ for all $s \in \closedint {-1} 1$.

From Probability Generating Function as Expectation the radius of convergence of both $\map {\Pi_X} s$ and $\Pi_Y \left({s}\right)$ is at least $1$.

Therefore they have unique power series expansions about $s = 0$:

$\ds \map {\Pi_X} s = \sum_{n \mathop = 0}^\infty s^n \map \Pr {X = k}$
$\ds \map {\Pi_Y} s = \sum_{n \mathop = 0}^\infty s^n \map \Pr {Y = k}$

As $\map {\Pi_X} s = \map {\Pi_Y} s$, these two power series have identical coefficients.

Hence the result.

$\blacksquare$