Table of contents |
2 Properties 3 Examples 4 Joint probability-generating functions 5 Related concepts |
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:
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:
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 sums of independent random variables. If X1, X2, ..., Xn is a sequence of independent (and not necessarily identically distributed) random variables, and
Note that this is the n-fold product of the probability-generating function of a Bernoulli random variable with parameter p. Note that this is the r-fold product of the probabiltiy generating function of a geometric random variable.
The probability-generating function is occasionally called the z-transform of the probability mass function. It is an example of a generating function of a sequence (see formal power series).
Other generating functions of random variables include the moment-generating function and the characteristic function.Definition
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.Properties
Power series
Probabilities and expectations
Sums of independent random variables
then the probability-generating function, GS(z), is given by
Further, 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 and identically distributed with common probability-generating function GX, then
This last fact is useful in the study of Galton-Watson processes.Examples
Related concepts