Probability Generating Functions
Chapter 1
Foundations of Probability and Random Variables
An introduction to probability theory, random variables, expectation, variance, and probability generating functions.
Introduction
Probability theory is the mathematical language of uncertainty. It provides a rigorous framework for describing random phenomena, quantifying uncertainty, and making predictions based on observed or theoretical information.
The concepts introduced in this chapter form the mathematical foundation for the remainder of this book. In particular, they provide the background necessary for understanding Probability Generating Functions (PGFs), which are one of the most powerful tools for studying discrete probability distributions.
Learning Objectives
After studying this chapter, you should be able to:
- Define a sample space and an event.
- Explain the axioms of probability.
- Distinguish between discrete and continuous random variables.
- Compute probability mass functions.
- Calculate expectations and variances.
- Explain generating functions.
Chapter Outline
- Introduction to Probability Theory
- Sample Spaces and Events
- Probability Measures
- Random Variables
- Discrete Random Variables
- Continuous Random Variables
- Expectation and Moments
- Variance and Standard Deviation
- Generating Functions
- Introduction to Probability Generating Functions
Chapter Summary
Prerequisites
Readers should understand basic algebra and mathematical notation.
What's Next?
The next section introduces the foundations of probability theory.