Probability-generating function

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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.

Contents

Definition

Univariate case

If X is a discrete random variable taking values in the non-negative integers {0,1, ...}, then the probability generating function of X is defined as [1]

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 X = (X1,...,Xd) is a discrete random variable taking values 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 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 ,
  2. 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.
  3. The normalization of the probability mass function can be expressed in terms of the generating function by
    The expectation of is given by
    More generally, the factorial moment, of is given by
    So the variance of is given by
    Finally, the raw moment of X is given by
  4. 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:

where the are constant natural numbers, then the probability generating function is given by
.
and
.
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.
, where .
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.

Examples

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
, 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 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

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