Table of contents |

2 Properties 3 Examples 4 Joint probability-generating functions 5 Related concepts |

If *X* is a discrete random variable taking values on some subset of the non-negative integers, {\*0,1, ...*}, then the *probability-generating function* of *X* is defined as:

Probability-generating functions obey all the rules of power series with non-negative coefficients. In particular, since *G*(1-) = 1 (since the probabilities must sum to one), 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. (Note that *G*(1-) = lim_{z↑1}*G*(*z*).)

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

- The probability mass function of
*X*is recovered by taking derivatives of*G*:

- It follows from Property 1 that if we have two random variables
*X*and*Y*, and*G*_{X}=*G*_{Y}, then*f*_{X}=*f*_{Y}. That is, if*X*and*Y*have identical probability-generating functions, then they are identically distributed.

- The expectation of
*X*is given byMore generally, the

*k*th factorial moment, E(*X*(*X*− 1) ... (X − k + 1)), of*X*is given by

Probability-generating functions are particularly useful for dealing with sums of independent random variables. If *X*_{1}, *X*_{2}, ..., *X*_{n} is a sequence of independent (and not necessarily identically distributed) random variables, and

- The probability-generating function of a constant random variable, i.e. one with Pr(
*X*=*c*) = 1, is

- The probability-generating function of a binomial random variable, the number of successes in
*n*trials, with probability*p*of success in each trial, isNote that this is the

*n*-fold product of the probability-generating function of a Bernoulli random variable with parameter*p*.

- The probability-generating function of a negative binomial random variable, the number of trials required to obtain the
*r*th success with probabiltiy of success in each trial*p*, isNote that this is the

*r*-fold product of the probabiltiy generating function of a geometric random variable.

- The probability-generating function of a Poisson random variable with rate parameter λ is

The probability-generating function is occasionally called the z-transform of the probability mass function. It is an example of a generating function of a sequence (see formal power series).

Other generating functions of random variables include the moment-generating function and the characteristic function.