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.
If X is a discrete random variable taking values x in the non-negative integers {0,1, ...}, then the probability generating function of X is defined as [1]
is the probability mass function of
Note that the subscripted notations
are often used to emphasize that these pertain to a particular random variable
, and to its distribution.
The power series converges absolutely at least for all complex numbers
; the radius of convergence being often larger.
If X = (X1,...,Xd) is a discrete random variable taking values (x1, ..., xd) 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
max
{\displaystyle {\text{max}}\{|z_{1}|,...,|z_{d}|\}\leq 1.}
Probability generating functions obey all the rules of power series with non-negative coefficients.
, x approaching 1 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.
The following properties allow the derivation of various basic quantities related to
: Probability generating functions are particularly useful for dealing with functions of independent random variables.
are constant natural numbers, then the probability generating function is given by
This can be seen, using the law of total expectation, as follows:
This last fact is useful in the study of Galton–Watson processes and compound Poisson processes.
For identically distributed
s, this simplifies to the identity stated before, but the general case is sometimes useful to obtain a decomposition of
by means of generating functions.
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
can also be considered for continuous and other random variables.