Probability-generating function explained

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 x in the non-negative integers, then the probability generating function of X is defined as^[1]

$$G(z)\; =\; \backslash operatorname\; (z^X)\; =\; \backslash sum\_^\; p(x)\; z^x,$$where

is the probability mass function of . Note that the subscripted notations and 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 with ; the radius of convergence being often larger.

Multivariate case

If is a discrete random variable taking values in the -dimensional non-negative integer lattice, then the probability generating function of is defined as$$G(z)\; =\; G(z\_1,\backslash ldots,z\_d)\; =\; \backslash operatorname\backslash bigl\; (z\_1^\backslash cdots\; z\_d^\backslash bigr)\; =\; \backslash sum\_^p(x\_1,\backslash ldots,x\_d)\; z\_1^\; \backslash cdots\; z\_d^,$$where is the probability mass function of . 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,

, where , 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.

Probabilities and expectations

The following properties allow the derivation of various basic quantities related to

:

1. The probability mass function of

is recovered by taking derivatives of , $$p(k)\; =\; \backslash operatorname(X\; =\; k)\; =\; \backslash frac.$$

1. It follows from Property 1 that if random variables

and have probability-generating functions that are equal, , then . That is, if and have identical probability-generating functions, then they have identical distributions.

1. The normalization of the probability mass function can be expressed in terms of the generating function by $$\backslash operatorname[1]\; =\; G(1^-)\; =\; \backslash sum\_^\backslash infty\; p(i)\; =\; 1.$$ The expectation of

is given by $$\backslash operatorname[X]\; =\; G\text{'}(1^-).$$ More generally, the factorial moment, of is given by $$\backslash operatorname\backslash left[\backslash frac\{X!\}\{(X-k)!\}\backslash right]\; =\; G^(1^-),\; \backslash quad\; k\; \backslash geq\; 0.$$ So the variance of is given by $$\backslash operatorname(X)=G$$(1^-) + G'(1^-) - \left [G'(1^-)\right ]^2. Finally, the -th raw moment of X is given by $$\backslash operatorname[X^k]\; =\; \backslash left(z\backslash frac\backslash right)^k\; G(z)\; \backslash Big|\_$$ where X is a random variable, is the probability generating function (of ) and is the moment-generating function (of ).

Functions of independent random variables

Probability generating functions are particularly useful for dealing with functions of independent random variables. For example:

Examples

• The probability generating function of an almost surely constant random variable, i.e. one with

and is $$G(z)\; =\; z^c.$$

• The probability generating function of a binomial random variable, the number of successes in

trials, with probability of success in each trial, is $$G(z)\; =\; \backslash left[(1-p)\; +\; pz\backslash right]^n.$$ Note: it is the -fold product of the probability generating function of a Bernoulli random variable with parameter . So the probability generating function of a fair coin, is $$G(z)\; =\; \backslash frac\; +\; \backslash frac.$$

• The probability generating function of a negative binomial random variable on

, the number of failures until the success with probability of success in each trial , is $$G(z)\; =\; \backslash left(\backslash frac\backslash right)^r,$$ which converges for . Note that this is the -fold product of the probability generating function of a geometric random variable with parameter on .

• The probability generating function of a Poisson random variable with rate parameter

is $$G(z)\; =\; e^.$$

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

can also be considered for continuous and other random variables.

Notes

1. Book: Probability and Distribution Theory . Gleb Gribakin .

References

• Book: Johnson . Norman Lloyd . Kotz . Samuel . Kemp . Adrienne W. . Adrienne W. Kemp. Univariate Discrete Distributions . 1992 . J. Wiley & Sons . 978-0-471-54897-3 . 2nd . Wiley series in probability and mathematical statistics . New York .