If X is a discrete random variable taking values in the non-negative integers, then the probability generating function of X is defined as where p is the probability mass function of X. Note that the subscripted notations GX and pX 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 numbersz with |z| ≤ 1; in many examples the radius of convergence is larger.
Multivariate case
If is a discrete random variable taking values in the d-dimensional non-negative integer lattice d, then the probability generating function of X is defined as where p is the probability mass function of X. The power series converges absolutely at least for all complex vectors with.
Properties
Power series
Probability generating functions obey all the rules of power series with non-negative coefficients. In particular, G = 1, where G = limz→1Gfrom 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:
The probability mass function of X is recovered by taking derivatives of G,
:
It follows from Property 1 that if random variablesX and Y have probability-generating functions that are equal,, then. That is, if X and Y have identical probability-generating functions, then they have identical distributions.
Probability generating functions are particularly useful for dealing with functions of independent random variables. For example:
If X1, X2,..., XN is a sequence of independent random variables, and
Suppose that N is also an independent, discrete random variable taking values on the non-negative integers, with probability generating function GN. If the X1, X2,..., XN are independent andidentically distributed with common probability generating function GX, then
Suppose again that N is also an independent, discrete random variable taking values on the non-negative integers, with probability generating function GN and probability density. If the X1, X2,..., XN are independent, but not identically distributed random variables, where denotes the probability generating function of, then
The probability generating function of a binomial random variable, the number of successes in n trials, with probability p of success in each trial, is
The probability generating function is an example of a generating function of a sequence: see alsoformal power series. It is equivalent to, and sometimes called, the z-transform of the probability mass function. Other generating functions of random variables include the moment-generating function, the characteristic function and the cumulant generating function. The probability generating function is also equivalent to the factorial moment generating function, which as can also be considered for continuous and other random variables.
