Probability-generating function

In probability theory, the probability generating function of a discrete random variable is a power series representation 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:

$G(z)={\textrm {E}}(z^{X})=\sum _{i=0}^{\infty }f(i)z^{i},$

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:

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

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

                 ${\textrm {E}}(X)=G'(1-).$

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

                 ${\textrm {E}}\left(X(X-1)\ldots (X-k+1)\right)=G^{(k)}(1-),\quad k\geq 1.$
    

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, then if

$S_{n}=\sum _{i=1}^{n}X_{i},$

the probability generating function, G_S(z), is given by

$G_{S_{n}}(z)=G_{X_{1}}(z)G_{X_{2}}(z)\ldots G_{X_{n}}(z).$

Further, if N is also a discrete random variable taking values on the non-negative integers, and the X₁, X₂, ..., X_N are independent and identically distributed with common probability generating function G_X, then

$G_{S_{N}}(z)=G_{N}(G_{X}(z)).$

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.