20.1.1 Probability Generating Functions (PGFs)

Introduction

A Probability Generating Function (PGF) provides a compact way to represent the probabilities of a discrete random variable. It is especially useful in studying the properties of random variables, such as the mean and variance, and in analysing distributions like the binomial, Poisson, geometric, and negative binomial distributions.

This note covers:

1. Definitions and derivations of PGFs.
2. Key properties of PGFs.
3. Applications to common distributions.

Definition of a Probability Generating Function

The Probability Generating Function (PGF) of a discrete random variable $X$ with probabilities $P(X = x) = p_x$ is defined as:

$$G_X(t) = \mathbb{E}[t^X] = \sum_{x=0}^\infty p_x t^x$$

where:

• $G_X(t)$ is the PGF of $X$
• $p_x = P(X = x)$ is the probability mass function (PMF) of $X$
• $t$ is a dummy variable (not necessarily a probability).

Properties of PGFs

Normalisation:

$$G_X(1) = \sum_{x=0}^\infty p_x = 1$$

Finding Probabilities:

The probability $P(X = x)$ can be obtained by differentiating $G_X(t)$ and evaluating at $t = 0$

$$P(X = x) = \frac{1}{x!} \frac{d^x}{dt^x} G_X(t) \bigg|_{t=0}$$

PGFs for Common Distributions

Binomial Distribution

For $X \sim B(n, p)$

$$G_X(t) = [pt + (1-p)]^n$$

Poisson Distribution

For $X \sim \text{Po}(\lambda)$

$$G_X(t) = e^{\lambda(t-1)}$$

Geometric Distribution

For $X \sim \text{Geom}(p)$ (number of failures before the first success):

$$G_X(t) = \frac{p}{1 - (1-p)t}, \quad |t| < \frac{1}{1-p}$$

Negative Binomial Distribution

For $X \sim \text{NegBin}(r, p)$ (number of failures before r successes):

$$G_X(t) = \left(\frac{p}{1 - (1-p)t}\right)^r, \quad |t| < \frac{1}{1-p}$$

Worked Example

Note Summary