Univariate case

If X is a discrete random variable taking values in the non-negative integers {0,1, ...}, then the probability generating function of X is defined as ^[1]

${\displaystyle G(z)=\operatorname {E} (z^{X})=\sum _{x=0}^{\infty }p(x)z^{x},}$

where p is the probability mass function of X. Note that the subscripted notations G_X and p_X are often used to emphasize that these pertain to a particular random variable X, and to its distribution. The power series converges absolutely at least for all complex numbers z with |z| ≤ 1; in many examples the radius of convergence is larger.

Multivariate case

If X = (X₁,...,X_d) is a discrete random variable taking values in the d-dimensional non-negative integer lattice {0,1, ...}^d, then the probability generating function of X is defined as

${\displaystyle G(z)=G(z_{1},\ldots ,z_{d})=\operatorname {E} {\bigl (}z_{1}^{X_{1}}\cdots z_{d}^{X_{d}}{\bigr )}=\sum _{x_{1},\ldots ,x_{d}=0}^{\infty }p(x_{1},\ldots ,x_{d})z_{1}^{x_{1}}\cdots z_{d}^{x_{d}},}$

where p is the probability mass function of X. The power series converges absolutely at least for all complex vectors ${\displaystyle z=(z_{1},...z_{d})\in \mathbb {C} ^{d}}$ with ${\displaystyle {\text{max}}\{|z_{1}|,...,|z_{d}|\}\leq 1.}$

Power series

Probability generating functions obey all the rules of power series with non-negative coefficients. In particular, G(1⁻) = 1, where G(1⁻) = lim_z→1G(z) from below, since the probabilities must sum to one. So the radius of convergence of any probability generating function must be at least 1, by Abel's theorem for power series with non-negative coefficients.

Probabilities and expectations

The following properties allow the derivation of various basic quantities related to X:

1. The probability mass function of X is recovered by taking derivatives of G,

   ${\displaystyle p(k)=\operatorname {Pr} (X=k)={\frac {G^{(k)}(0)}{k!}}.}$

2. It follows from Property 1 that if random variables X and Y have probability-generating functions that are equal, ${\displaystyle G_{X}=G_{Y}}$, then ${\displaystyle p_{X}=p_{Y}}$. That is, if X and Y have identical probability-generating functions, then they have identical distributions.
3. The normalization of the probability density function can be expressed in terms of the generating function by

   ${\displaystyle \operatorname {E} [1]=G(1^{-})=\sum _{i=0}^{\infty }p(i)=1.}$

   The expectation of ${\displaystyle X}$ is given by

   ${\displaystyle \operatorname {E} [X]=G'(1^{-}).}$

   More generally, the k^th factorial moment, ${\displaystyle \operatorname {E} (X(X-1)\cdots (X-k+1))}$ of X is given by

   ${\displaystyle \operatorname {E} \left[{\frac {X!}{(X-k)!}}\right]=G^{(k)}(1^{-}),\quad k\geq 0.}$

   So the variance of X is given by

   ${\displaystyle \operatorname {Var} (X)=G''(1^{-})+G'(1^{-})-\left[G'(1^{-})\right]^{2}.}$

   Finally, the k^th raw moment of X is given by

   ${\displaystyle \operatorname {E} [X^{k}]=\left(z{\frac {\partial }{\partial z}}\right)^{k}G(z){\Big |}_{z=1^{-}}}$

4. ${\displaystyle G_{X}(e^{t})=M_{X}(t)}$ where X is a random variable, ${\displaystyle G_{X}(t)}$ is the probability generating function (of X) and ${\displaystyle M_{X}(t)}$ is the moment-generating function (of X) .

Functions of independent random variables

Probability generating functions are particularly useful for dealing with functions of independent random variables. For example:

• If X₁, X₂, ..., X_N is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and

${\displaystyle S_{N}=\sum _{i=1}^{N}a_{i}X_{i},}$

where the a_i are constant natural numbers, then the probability generating function is given by

${\displaystyle G_{S_{N}}(z)=\operatorname {E} (z^{S_{N}})=\operatorname {E} \left(z^{\sum _{i=1}^{N}a_{i}X_{i},}\right)=G_{X_{1}}(z^{a_{1}})G_{X_{2}}(z^{a_{2}})\cdots G_{X_{N}}(z^{a_{N}}).}$

For example, if

${\displaystyle S_{N}=\sum _{i=1}^{N}X_{i},}$

then the probability generating function, G_SN(z), is given by

${\displaystyle G_{S_{N}}(z)=G_{X_{1}}(z)G_{X_{2}}(z)\cdots G_{X_{N}}(z).}$

It also follows that the probability generating function of the difference of two independent random variables S = X₁ − X₂ is

${\displaystyle G_{S}(z)=G_{X_{1}}(z)G_{X_{2}}(1/z).}$

• Suppose that N, the number of independent random variables in the sum above, is not a fixed natural number but is also an independent, discrete random variable taking values on the non-negative integers, with probability generating function G_N. If the X₁, X₂, ..., X_N are independent and identically distributed with common probability generating function G_X, then

${\displaystyle G_{S_{N}}(z)=G_{N}(G_{X}(z)).}$

This can be seen, using the law of total expectation, as follows:

${\displaystyle {\begin{aligned}G_{S_{N}}(z)&=\operatorname {E} (z^{S_{N}})=\operatorname {E} (z^{\sum _{i=1}^{N}X_{i}})\\[4pt]&=\operatorname {E} {\big (}\operatorname {E} (z^{\sum _{i=1}^{N}X_{i}}\mid N){\big )}=\operatorname {E} {\big (}(G_{X}(z))^{N}{\big )}=G_{N}(G_{X}(z)).\end{aligned}}}$

This last fact is useful in the study of Galton–Watson processes and compound Poisson processes.

• Suppose again that N is also an independent, discrete random variable taking values on the non-negative integers, with probability generating function G_N and probability mass function ${\displaystyle f_{i}=\Pr\{N=i\}}$. If the X₁, X₂, ..., X_N are independent, but not identically distributed random variables, where ${\displaystyle G_{X_{i}}}$ denotes the probability generating function of ${\displaystyle X_{i}}$, then

${\displaystyle G_{S_{N}}(z)=\sum _{i\geq 1}f_{i}\prod _{k=1}^{i}G_{X_{i}}(z).}$

For identically distributed X_i this simplifies to the identity stated before. The general case is sometimes useful to obtain a decomposition of S_N by means of generating functions.