Definition:Probability Generating Function

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.