Chapter Four

Random Variables: The Discrete Case

# 4.1 Introduction/Purpose of the Chapter

This chapter treats discrete random variables. After having introduced the general notion of a random variable, we discuss specific cases. Discrete random variables are presented next, and continuous random variables are left to the next chapter. In this chapter we learn about calculating simple probabilities using a probability mass function. Several probability functions for discrete random variables warrant special mention because they arise frequently in real-life situations. These are the probability functions for, among others, the so-called geometric, hypergeometric, binomial, and Poisson distributions. We focus on the physical assumptions underlying the application of these functions to real problems. Although we can use computers to calculate probabilities from these distributions, it is often convenient to use special tables, or even use approximate methods in which one probability function can be approximated quite closely by another function. We introduce the concepts of distribution, cumulative distribution function, expectation, and variance for discrete random variables. We also discuss higher-order moments of such variables.

# 4.2 Vignette/Historical Notes

Historically, the discrete random variables were the first type of random outcomes studied in practice. The documented exchange of letters in 1964 between Pascal and Fermat was prompted by a game of dice which essentially dealt ...