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 $\map {\Pi_X} s$, 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.

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Also known as

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

Some prefer PGFof $X$ to PGFfor $X$.

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:

$\ds \map {\Pi_X} s := \sum_{n \mathop \in \Omega_X} \map {p_X} n s^n$

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

Yet others define $\map {\Pi_X} s := \expect {s^X}$, 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 $\expect {s^X}$ 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
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): probability generating function