Probability-generating function

In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr("X" = "i") in the probability mass function for a random variable "X", and to make available the well-developed theory of power series with non-negative coefficients.
Definition.
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
where "p" is the probability mass function of "X". Note that the subscripted notations "G""X" 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 numbers "z" 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 {0,1, ...}"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−) = 1, where "G"(1−) = limz→1"G"("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":
Functions of independent random variables.
Probability generating functions are particularly useful for dealing with functions of independent random variables. For example:
Related concepts.
The probability generating function is an example of a generating function of a sequence: see also formal 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 formula_33 can also be considered for continuous and other random variables.