Chapter 2: Discrete Probability Distributions and Generating Functions
Overview
This chapter introduces the mathematical foundations of discrete probability distributions and their connection to generating functions.
Probability Generating Functions (PGFs) provide a powerful method for representing discrete probability distributions and analysing their properties.
In this chapter, we study:
- Discrete random variables.
- Probability mass functions.
- Cumulative distribution functions.
- Expected values and moments.
- Generating functions.
- Probability Generating Functions.
Chapter Introduction
Discrete probability distributions describe random systems where outcomes can be counted.
Probability Generating Functions transform these probability models into algebraic forms that allow easier analysis of probabilities, moments, and relationships between random variables.
Chapter Objectives
After completing this chapter, the reader should be able to:
- Define discrete random variables.
- Construct probability mass functions.
- Calculate probabilities from discrete distributions.
- Understand cumulative distribution functions.
- Introduce generating functions.
- Construct Probability Generating Functions.
- Calculate moments using PGFs.
Chapter Structure
2.1 Discrete Random Variables
A discrete random variable takes values from a countable set.
Examples include:
- Number of customers arriving at a service station.
- Number of failures in a system.
- Number of infected individuals in a population.
- Number of insurance claims.
A discrete random variable can take values: