# Definition:Probability Generating Function

## Contents

## Definition

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 $\Pi_X \left({s}\right)$, is the formal power series defined by:

- $\displaystyle \Pi_X \left({s}\right) := \sum_{n \mathop = 0}^\infty p_X \left({n}\right) s^n \in \R \left[\left[{s}\right]\right]$

that is, it is the generating function associated with the sequence $\left\langle{p_X \left({n}\right)}\right\rangle_{n \in \N}$.

Note that in the case that $\Omega_X$ is finite, $\Pi_X$ is in fact a polynomial in $\R \left[{s}\right]$, since then only finitely many $p_X \left({n}\right)$ are nonzero.

## Also known as

The term **probability generating function** is sometimes (conveniently) abbreviated to one of **p.g.f.**, **P.G.F.** or simply **PGF**.

Some prefer **PGF of $X$** to

**PGF**.

*for*$X$

There are also various different notations in use for $\Pi_X$, like $G_X$ or $U_X$.

If $X$ is clear from the context $\Pi$ is often used to enhance simplicity and readability.

Further, some sources prefer to write the defining formal summation above as:

- $\displaystyle \Pi_X \left({s}\right) := \sum_{n \mathop \in \Omega_X} p_X \left({n}\right) s^n$

but this comes down to the same thing by definition of the probability mass function $p_X$.

Yet others define $\Pi_X \left({s}\right) := E \left({s^X}\right)$, with the latter being the expectation of $s^X$.

Although intuitively correct, this approach is to be discouraged, since one needs $s^X$ to be a random variable in order for $E \left({s^X}\right)$ to make sense.

But $s$ is still a formal variable, and no assertions on convergence have been made at this point.

Therefore, to prevent the uninitiated from glossing over convergence issues, it is didactically preferable not to introduce the **PGF** via this route.

The identification is formally justified in Probability Generating Function as Expectation.

## Also see

- Results about
**probability generating functions**can be found here.

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 4.2$: Integer-valued random variables