The probabilitygenerating function of a discrete random variable is a power series representation of the probability mass function of the random variable.
Ifis a discrete random variable taking values in the nonnegative integersthen the probabilitygenerating function ofis defined as
whereis the probability mass function of
If is a discrete random variable taking values in the ddimensional nonnegative integer latticethen the probabilitygenerating function ofis defined as
whereis the probability mass function of X. The power series converges absolutely for all complex vectorswith
Probability generating functions obey all the rules of power series with nonnegative coefficients.
The following properties allow the derivation of various basic quantities related to
1. The probability mass function ofis recovered by taking derivatives of
2. If two probability distributions has the same probability generating function then they are the same distribution.
The expectation ofis given bywhereindicatesfrom below.
More generally, the kth factorial moment,of X is given bySo the variance of X is given by
3.whereis a random variable,is the probability generating function andis the momentgenerating function.
Probabilitygenerating functions are useful when several independent random variables are involved. For example:

Ifis a sequence of independent random variables, and
 where theare constants, then the probabilitygenerating function is given by
 For example, if Sn =SUM^n{i=1} Xi then
 It follows that the probabilitygenerating function of the difference of two independent random variablesis
Example:

The probabilitygenerating function of a constant random variable, i.e. one withis

The probabilitygenerating function of a binomial random variable, the number of successes intrials, with probabilityof success in each trial, is

The probabilitygenerating function of a negative binomial random variable, the number of failures occurring before theth success with probability of success in each trialis
 Note that this is theth power of the probability generating function of a geometric random variable.

The probabilitygenerating function of a Poisson random variable with rate parameteris