\begin{definition}[Definition:Probability Generating Function]
Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$.
Let $p_X$ be the probability mass function for $X$.
The '''probability generating function for $X$''', denoted $\map {\Pi_X} s$, is the formal power series defined by:
:$\ds \map {\Pi_X} s := \sum_{n \mathop = 0}^\infty \map {p_X} n s^n \in \R \sqbrk {\sqbrk s}$
that is, it is the generating function associated with the sequence $\sequence {\map {p_X} n}_{n \mathop \in \N}$.
Note that in the case that $\Omega_X$ is finite, $\Pi_X$ is in fact a polynomial in $\R \sqbrk s$, since then only finitely many $\map {p_X} n$ are non-zero.
{{explain|Discrepancy between notations: $\R \sqbrk {\sqbrk s}$ vs. $\R \sqbrk s$, neither of which is formally defined.}}
\end{definition}