Discrete Random Variables

Summary :

This chapter uses business examples, including the number of long-distance calls employees make during peak hours, to explain discrete random variables and probability density functions. It covers random variable notation, the two defining properties of a probability density function, and works through the hypergeometric, binomial, and geometric distributions as the main discrete models covered.

Random variables and their notation

A discrete random variable takes only countable, whole-number values, and the chapter establishes the convention of an upper-case letter such as X to denote the random variable in words and a lower-case letter such as x for a specific numeric outcome, illustrated with counting the number of heads in three coin tosses.

Properties of a probability density function

A probability density function for a discrete random variable must satisfy two conditions: each individual probability lies between zero and one inclusive, and the sum of all the probabilities across every possible outcome equals one, a check used throughout the chapter to confirm that a proposed distribution is valid.

Hypergeometric, binomial, and geometric distributions

The chapter develops three named discrete distributions in turn: the hypergeometric distribution, presented as the simplest probability density function for sampling without replacement from a finite population; the binomial distribution, described as a more broadly applicable model with many business uses; and the geometric distribution, which builds on the binomial setup to model the number of trials until a first success.