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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:
where f is the probability mass function of X. Note that the equivalent notation GX is sometimes used to distinguish between the probability-generating functions of several random variables.
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-) = limz↑1G(z).)
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
2. It follows from Property 1 that if we have two random variables X and Y, and GX = GY, then fX = fY. That is, if X and Y have identical probability-generating functions, then they are identically distributed.
3. The normalization of the probability density function can be expressed in terms of the generating function by
The expectation of X is given by
More generally, the kth factorial moment, E(X(X − 1) ... (X − k + 1)), of X is given by
Probability-generating functions are particularly useful for dealing with functions of independent random variables. For example: