Probability Generating Function
For a random variable taking values in , the probability generating function is
where .
Let be a discrete random variable taking values in . The probability generating function (PGF) of is defined by
where .
Convergence
Since and , the series converges absolutely for . The radius of convergence of is at least , so is well-defined on the closed unit disc.
The PGF determines the distribution
Proof of uniqueness
Suppose and are two probability distributions with the same PGF, so for all ,
The coefficients agree by induction. Letting gives . Assuming for all , subtracting the common part leaves
and dividing by and letting gives .
Sums of independent random variables
If are independent random variables with PGFs , the PGF of is the product:
For example, if and are independent, then
so ; and if and are independent, then , so .
Derivative at one gives the expectation
Let be the PGF of a random variable . Then
Proof of the derivative formula
First suppose . For ,
so is increasing in and bounded above by , giving . Conversely, for choose with ; then
and since is arbitrary the two bounds agree.
If , then for every there is an with , and the same partial-sum bound gives , so .
Higher derivatives and moments
Let be the PGF of a random variable . Then
Examples
For ,
For ,
For ,
Related
Stated in
- Definition 3.56 (Probability Generating Function)§3.6.1 Introduction
- Theorem 3.57§3.6.1 Introduction
- Theorem 3.63§3.6.1 Introduction
- Theorem 3.64§3.6.1 Introduction