## Quick Reference

(pgf)

For the discrete random variable *X*, with probability distribution P(*X*=*x** _{j}*),

*j*=1, 2, 3,…, the probability-generating function G is defined by where

*t*is an arbitrary variable. Note that G(

*t*) is the expectation of

*t*

*and G(1)=1. If the set of possible values of*

^{X}*x*is infinite, |

*t*| needs to be small enough for the series to converge.

If the first and second derivatives of G(*t*) with respect to *t* are denoted by G′(*t*) and G″(*t*), respectively, the expectation and variance of *X* are given by G′(1) and G″(1)+G′(1)−{G′(1)}^{2}, where, for example, G′(1) denotes the value of G′(*t*) when *t*=1.

Like the moment-generating function the probability-generating function can provide a useful alternative description of a probability distribution. For example, if *Y* denotes the sum of *n* independent random variables, each having pgf G(*t*), then P(*Y*=*y*) is the coefficient of *t** ^{y}* in {G(

*t*)}

*.*

^{n}Another useful property is that, if *X* and *Y* are independent random variables with probability-generating functions G* _{X}*(

*t*) and G

*(*

_{Y}*t*), respectively, then the probability-generating function of

*Z*=

*X*+

*Y*is G

*(*

_{Z}*t*), where

G* _{Z}*(

*t*)=G

*(*

_{X}*t*)×G

*(*

_{Y}*t*).

De Moivre used the probability-generating function technique in 1730. The term itself became common following its use by Bartlett in 1940. See also moment-generating function.

*Subjects:*
Probability and Statistics.