5.1 Introduction: Discrete Random Variables

This photo shows branch lightening coming from a dark cloud and hitting the ground.
You can use probability and discrete random variables to calculate the likelihood of lightning striking the ground five times during a half-hour thunderstorm. (Credit: Leszek Leszczynski)

Learning Objectives

By the end of this chapter, the student should be able to:

  • Define a a random variable for an experiment.
  • Recognize, create, and use a discrete probability distribution function.
  • Calculate and interpret expected values of a discrete probability distribution.
  • Calculate and interpret the standard deviation of a discrete probability distribution.
  • Recognize a binomial random variable and create a binomial probability distribution function.
  • Recognize a Poisson random variable and create a Poisson probability distribution function.
In your math background, you have used variables. At its core, a variable is a quantity that can change. In a typical algebra class we would let letters like x and y represent variables. In statistics, we will use a similar idea when we talk about a random variable.

Random Variable

A random variable takes on value as the result of an experiment.

Random variables can be discrete or continuous and both will be explored. For now, we should discuss the common notation we use in statistics to represent a random variable as well as the values the random variable can take on.

Random Variable Notation

Upper case letters such as X or Y denote the meaning or the definition of a random variable. Lower case letters like x or y denote the value of a random variable. If X  is a random variable, then X  is given a written description in words, and x is given as a number. A random variable represents a characteristic of interest in a population being studied and takes on a numeric value as the result of an experiment. For example, if the random variable X is the number of children in a family, then x represents a specific integer 0, 1, 2, 3, etc.

Random variables in statistics differ from variables in an algebra class in the two following ways.

  • The domain of the random variable is not necessarily a numerical set; the domain may be expressed in words; for example, if X = hair color then the domain is {black, blond, gray, green, orange}.
  • We can tell what specific value x the random variable X takes only after performing the experiment.

For example, let X  represent the number of heads you get when you toss three fair coins. Notice the upper case X is described in words and specifies a future experiment. The sample space for the toss of three fair coins consists of 8 elements and can be listed as TTT, THH, HTH, HHT, HTT, THT, TTH, HHH. For each outcome in the sample space, when the number of heads are counted, there are only four possible values the random variable can take on, namely, x = 0, 1, 2, 3. Notice that for this example, the x values are countable outcomes. Because we can count the possible values that the variable X can take on and the outcomes are random, X is called a discrete random variable. In this section, we will discuss discrete random variables.

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Introduction to Statistics for Engineers Copyright © by Vikki Maurer & Jeff Crabill & Linn-Benton Community College is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.