Probability Generating Functions and Biological Applications

This online text is currently a work in progress intended for a Jan 2026 AMSI summer school class on Probability Generating Functions (PGFs) in Biological Applications.

I have spent a lot of time thinking about the structure of this book. The goal is to simultaneously learn about PGFs and the biological applications. It is natural to learn them in parallel. So I considered weaving the mathematics and applications together so that the sections would alternate between mathematics and biology.

I’ve decided not to do this (at least for now). I am placing the mathematical properties in a separate part of the book from the biological applications. However, I will recommend an order for the reading which will move between the two parts. The Biological Application sections will reference which of the mathematical sections you are expected to have read.

Table of Contents

Foreword

• Prerequisites
• List of Theorems

Mathematical Properties of Generating Functions

• 1. Generating Functions

Biological Applications of Generating Functions

• 2. The Galton-Watson Process
• 3. Continuous-Time Galton-Watson Processes
• 4. Infectious Diseases
• 5. Introduction to MultiType Galton-Watson Processes

Simulations and Practice Problems

• 6. Computer Simulation
• 7. Practice Classes
• 8. Assignments