Chapter 3: Theory and Mathematical Properties of Probability Generating Functions

Chapter Introduction

In the previous chapter, Probability Generating Functions (PGFs) were introduced as algebraic representations of discrete probability distributions.

We constructed PGFs for important distributions including:

• Bernoulli distributions.
• Binomial distributions.
• Geometric distributions.
• Negative Binomial distributions.
• Poisson distributions.

This chapter develops the mathematical theory underlying Probability Generating Functions.

Rather than focusing only on computation, we establish the fundamental properties that make PGFs powerful analytical tools in probability theory.

Beginning with the mathematical structure of a PGF as an infinite power series, we study:

• Normalization.
• Probability recovery.
• Uniqueness.
• Convergence.
• Moment generation.

These results demonstrate that a PGF completely characterizes the probability distribution of a discrete random variable.

The chapter also studies how PGFs behave under addition of independent random variables and develops the composition rule for compound distributions.

These properties provide the mathematical foundation for applications in:

• Branching processes.
• Queueing theory.
• Actuarial mathematics.
• Reliability theory.
• Stochastic modelling.

Learning Objectives

After completing this chapter, the reader should be able to:

• Understand the mathematical structure of a Probability Generating Function.
• Study the fundamental properties satisfied by every PGF.
• Prove the normalization property G(1)=1.
• Show that G(0) represents P(X=0).
• Recover probabilities from derivatives of a PGF.
• Understand the uniqueness theorem for PGFs.
• Study convergence properties of PGFs.
• Derive expectations and factorial moments using differentiation.
• Obtain variance formulas using PGF derivatives.
• Understand multiplication of PGFs for independent sums.
• Study composition rules for compound distributions.
• Apply PGF theory to advanced stochastic models.

Chapter Structure

3.1 Introduction

Introduction to the mathematical role of Probability Generating Functions.

3.2 Mathematical Structure of a Probability Generating Function

Topics:

• Infinite power series representation.
• Coefficients as probabilities.
• Radius of convergence.

3.3 Fundamental Properties of Probability Generating Functions

Topics:

• Basic PGF properties.
• Non-negativity.
• Normalization.
• Uniqueness.

3.4 The Normalization Property

Proof: