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.

Contents

1 Definition 2 Properties 2.1 Power series 2.2 Probabilities and expectations 2.3 Functions of independent random variables 3 Examples 4 Related concepts

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:

<math>G(z) = \textrm{E}(z^X) = \sum_{i=0}^{\infty}f(i)z^i,<math>

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

<math> \quad f(k) = \textrm{Pr}(X = k) = \frac{G^{(k)}(0)}{k!}.<math>

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 normalization of the probability density function can be expressed in terms of the generating function by

<math> E(1)=G(1-)=\sum_{i=0}^\infty f(i)=1.<math>

The expectation of X is given by

<math> \textrm{E}\left(X\right) = G'(1-).<math>

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

<math>\textrm{E}\left(\frac{X!}{(X-k)!}\right) = G^{(k)}(1-), \quad k \geq 0.<math>

So we can get the variance of X as

<math>var(X)=G''(1-) + G'(1-) - \left [G'(1-)\right ]^2.<math>

Functions of independent random variables

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

• If X₁, X₂, ..., X_n is a sequence of independent (and not necessarily identically distributed) random variables, and

<math>S_n = \sum_{i=1}^n a_i X_i,<math>

where the a_i are constants, then the probability-generating function is given by

<math>G_{S_n}(z) = G_{X_1}(z^{a_1})G_{X_2}(z^{a_2})\ldots G_{X_n}(z^{a_n}).<math>

For example, if

<math>S_n = \sum_{i=1}^n X_i,<math>

then the probability-generating function, G_Sn(z), is given by

<math>G_{S_n}(z) = G_{X_1}(z)G_{X_2}(z)\ldots G_{X_n}(z).<math>

It also follows that the probability-generating function of the difference of two random variables S = X₁ − X₂ is

<math>G_S(z) = G_{X_1}(z)G_{X_2}(1/z).<math>

• 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

<math>G_{S_N}(z) = G_N(G_X(z)).<math>

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

<math>G(z) = \left(z^c\right).<math>

• 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

<math>G(z) = \left[(1-p) + pz\right]^n.<math>

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 probability of success in each trial p, is

<math>G(z) = \left(\frac{pz}{1 - (1-p)z}\right)^r.<math>

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

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

<math>G(z) = \left(\textrm{e}^{\lambda(z - 1)}\right).<math>

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.