Probability-generating function

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), and to make available the well-developed theory of power series with non-negative coefficients.

Definition

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:

where f is the probability mass function of X. Note that the equivalent notation G_X is sometimes used to distinguish between the probability-generating functions of several random variables.

Properties

Power series

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-) = lim_z↑1G(z).)

Probabilities and expectations

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

1. The probability mass function of X is recovered by taking derivatives of G:

                  
    

2. It follows from Property 1 that if we have two random variables X and Y, and G_X = G_Y, then f_X = f_Y. That is, if X and Y have identical probability-generating functions, then they are identically distributed.
    
3. The expectation of X is given by

                 E(X) = G'(1 - ).

   More generally, the kth factorial moment, E(X(X − 1) ... (X − k + 1)), of X is given by

                  
    

Sums of independent random variables

Probability-generating functions are particularly useful for dealing with sums of independent random variables. If X₁, X₂, ..., X_n is a sequence of independent (and not necessarily identically distributed) random variables, and

then the probability-generating function, G_S(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 G_N. If the X₁, X₂, ..., X_N are independent and identically distributed with common probability-generating function G_X, then

This last fact is useful in the study of Galton-Watson processes.

Examples

• The probability-generating function of a constant random variable, i.e. one with Pr(X = c) = 1, is

                G(z) = z^c.
   

• The probability-generating function of a binomial random variable, the number of successes in n trials, with probability p of success in each trial, is

                 

  Note that this is the n-fold product of the probability-generating function of a Bernoulli random variable with parameter p.
   

• The probability-generating function of a negative binomial random variable, the number of trials required to obtain the rth success with probabiltiy of success in each trial p, is

                 

  Note that this is the r-fold product of the probabiltiy generating function of a geometric random variable.
   

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

                G(z) = e^{λ(z - 1)}.

Related concepts

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.