Probability Generating Function defines Probability Distribution

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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 $\Pi_X \left({s}\right)$ and $\Pi_Y \left({s}\right)$ respectively.


Then:

$\forall s \in \left[{-1 \,.\,.\, 1}\right]: \Pi_X \left({s}\right) = \Pi_Y \left({s}\right)$

if and only if:

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


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


Proof

By the definition of PGF, it follows that if:

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

then:

$\forall s \in \left[{-1 \,.\,.\, 1}\right]: \Pi_X \left({s}\right) = \Pi_Y \left({s}\right)$


Suppose that $\Pi_X \left({s}\right) = \Pi_Y \left({s}\right)$ for all $s \in \left[{-1 \,.\,.\, 1}\right]$.

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

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

$\displaystyle \Pi_X \left({s}\right) = \sum_{n \mathop = 0}^\infty s^n \Pr \left({X = k}\right)$
$\displaystyle \Pi_Y \left({s}\right) = \sum_{n \mathop = 0}^\infty s^n \Pr \left({Y = k}\right)$

As $\Pi_X \left({s}\right) = \Pi_Y \left({s}\right)$, these two power series have identical coefficients.


Hence the result.

$\blacksquare$


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