4.1 Two Types of Random Variables
Recall that the sample-point probabilities corresponding to an experiment must sum to 1. Dividing one unit of probability among the sample points in a sample space and, consequently, assigning probabilities to the values of a random variable are not always as easy as the examples in Chapter 3 might lead you to might believe. If the number of sample points can be completely listed, the job is straightforward. But if the experiment results in an infinite number of sample points, which is impossible to list, the task of assigning probabilities to the sample points is impossible without the aid of a probability model. The next three examples demonstrate the need for different probability models, depending on the number of values that a random variable can assume.
Example 4.1 Values of a Discrete Random Variable—Wine Ratings
Problem
A panel of 10 experts for the Wine Spectator (a national publication) is asked to taste a new white wine and assign it a rating of 0, 1, 2, or 3. A score is then obtained by adding together the ratings of the 10 experts. How many values can this random variable assume?
Solution
A sample point is a sequence of 10 numbers associated with the rating of each expert. For example, one sample point is
[&|cbo|1, 0, 0, 1, 2, 0, 0, 3, 1, 0|cbc| &]
[Note: Using one of thea counting rules from Section 3.7, there are a total of sample points for this experiment.] Now, the random variable assigns a score to each one of these sample points by adding the 10 numbers together. Thus, the smallest score is 0 (if all 10 ratings are 0), and the largest score is 30 (if all 10 ratings are 3). Since every integer between 0 and 30 is a possible score, the random variable denoted by the symbol x can assume 31 values. Note that the value of the random variable for the sample point shown above is *
Look Back
Use of the random variable reduces the amount of information required in the experiment—from over 1 million sample points to 31 values of the random variable. Also, this is an example of a discrete random variable, since there is a finite number of distinct possible values. Whenever all the possible values that a random variable can assume can be listed (or counted), the random variable is discrete.
Example 4.2 Values of a Discrete Random Variable—EPA Application
Problem
Suppose the Environmental Protection Agency (EPA) takes readings once a month on the amount of pesticide in the discharge water of a chemical company. If the amount of pesticide exceeds the maximum level set by the EPA, the company is forced to take corrective action and may be subject to penalty. Consider the random variable x to be the number of months before the company’s discharge exceeds the EPA’s maximum level. What values can x assume?
Solution
The company’s discharge of pesticide may exceed the maximum allowable level on the first month of testing, the second month of testing, etc. It is possible that the company’s discharge will never exceed the maximum level. Thus, the set of possible values for the number of months until the level is first exceeded is the set of all positive integers
Look Back
If we can list the values of a random variable x, even though the list is never ending, we call the list countable and the corresponding random variable discrete. Thus, the number of months until the company’s discharge first exceeds the limit is a discrete random variable.
Now Work Exercise 4.7
Example 4.3 Values of a Continuous Random Variable—Another EPA Application
Problem
Refer to Example 4.2. A second random variable of interest is the amount x of pesticide (in milligrams per liter) found in the monthly sample of discharge waters from the same chemical company. What values can this random variable assume?
Solution
Some possible values of x are 1.7, 28.42, and 100.987 milligrams per liter. Unlike the number of months before the company’s discharge exceeds the EPA’s maximum level, the set of all possible values for the amount of discharge cannot be listed (i.e., is not countable). The possible values for the amount x of pesticide would correspond to the points on the interval between 0 and the largest possible value the amount of the discharge could attain, the maximum number of milligrams that could occupy 1 liter of volume. (Practically, the interval would be much smaller, say, between 0 and 500 milligrams per liter.)
Look Ahead
When the values of a random variable are not countable but instead correspond to the points on some interval, we call the variable a continuous random variable. Thus, the amount of pesticide in the chemical plant’s discharge waters is a continuous random variable.
Now Work Exercise 4.9
Random variables that can assume a countable number of values are called discrete.
Random variables that can assume values corresponding to any of the points contained in an interval are called continuous.
The following are examples of discrete random variables:
The number of seizures an epileptic patient has in a given week:
The number of voters in a sample of 500 who favor impeachment of the president:
The shoe size of a tennis player:
The change received for paying a bill:
The number of customers waiting to be served in a restaurant at a particular time:
Note that several of the examples of discrete random variables begin with the words The number of. … This wording is very common, since the discrete random variables most frequently observed are counts. The following are examples of continuous random variables:
The length of time (in seconds) between arrivals at a hospital clinic: (infinity)
The length of time (in minutes) it takes a student to complete a one-hour exam:
The amount (in ounces) of carbonated beverage loaded into a 12-ounce can in a can-filling operation:
The depth (in feet) at which a successful oil-drilling venture first strikes oil: where c is the maximum depth obtainable
The weight (in pounds) of a food item bought in a supermarket:
[Note: Theoretically, there is no upper limit on x, but it is unlikely that it would exceed 500 pounds.]
Discrete random variables and their probability distributions are discussed in Sections 4.2 and 4.3. Continuous random variables and their probability distributions are the topic of Sections 4.4 and 4.5.
Exercises 4.1–4.16
Understanding the Principles
4.1 What is a random variable?
4.2 How do discrete and continuous random variables differ?
Applying the Concepts—Basic
4.3 Type of Random Variable. Classify the following random variables according to whether they are discrete or continuous:
The number of words spelled correctly by a student on a spelling test
The amount of water flowing through the Hoover Dam in a day
The length of time an employee is late for work
The number of bacteria in a particular cubic centimeter of drinking water
The amount of carbon monoxide produced per gallon of unleaded gas
Your weight
4.4 Type of Random Variable. Identify the following random variables as discrete or continuous:
The amount of flu vaccine in a syringe
The heart rate (number of beats per minute) of an American male
The time it takes a student to complete an examination
The barometric pressure at a given location
The number of registered voters who vote in a national election
Your score on the either the SAT or ACT
4.5 Type of Random Variable. Identify the following variables as discrete or continuous:
The difference in reaction time to the same stimulus before and after training
The number of violent crimes committed per month in your community
The number of commercial aircraft near-misses per month
The number of winners each week in a state lottery
The number of free throws made per game by a basketball team
The distance traveled by a school bus each day
4.6 NHTSA crash tests. The National Highway Traffic Safety Administration (NHTSA) has developed a driver-side “star” scoring system for crash-testing new cars. Each crash-tested car is given a rating ranging from one star to five stars the more stars in the rating, the better is the level of crash protection in a head-on collision. Suppose that a car is selected and its driver-side star rating is determined. Let x equal the number of stars in the rating. Is x a discrete or continuous random variable?
4.7 Customers in line at a Subway shop. The number of customers, x, waiting in line to order sandwiches at a Subway shop at noon is of interest to the store manager. What values can x assume? Is x a discrete or continuous random variable? 4.8 Sound waves from a basketball. Refer to the American Journal of Physics (June 2010) experiment on sound waves produced from striking a basketball, Exercise 2.43 (p. 52). Recall that theThe frequencies of sound wave echoes resulting from striking a hanging basketball with a metal rod were recorded. Classify the random variable, frequency (measured in hertz) of an echo, as discrete or continuous.
- 4.9 Mongolian desert ants. Refer to Consider the Journal of Biogeography (Dec. 2003) study of ants in Mongolia, presented in Exercise 2.68 (p. 63). Two of the several variables recorded at each of 11 study sites were annual rainfall (in millimeters) and number of ant species. Identify these variables as discrete or continuous.
4.10 Motivation of drug dealers. Refer toConsider the Applied Psychology in Criminal Justice (Sept. 2009) study of the personality characteristics of drug dealers, Exercise 2.102 (p. 77). For each of 100 convicted drug dealers, the researchers measured several variables, including the number of prior felony arrests x. Is x a discrete or continuous random variable? Explain.
Applying the Concepts—Intermediate
4.11 Psychology. Give an example of a discrete random variable of interest to a psychologist.
4.12 Sociology. Give an example of a discrete random variable of interest to a sociologist.
4.13 Nursing. Give an example of a discrete random variable of interest to a hospital nurse.
4.14 Art history. Give an example of a discrete random variable of interest to an art historian.
4.15 Irrelevant speech effects. Refer toConsider the Acoustical Science & Technology (Vol. 35, 2014) study of the degree to which the memorization process is impaired by irrelevant background speech (called irrelevant speech effects), Exercise 2.34 (p. 49). Recall that subjectsSubjects performed a memorization task under two conditions: (1) with irrelevant background speech and (2) in silence. Let x represent the difference in the error rates for the two conditions—called the relative difference in error rate (RDER). Explain why x is a continuous random variable.
4.16 Shaft graves in ancient Greece. Refer toConsider the American Journal of Archaeology (Jan. 2014) study of shaft graves in ancient Greece, Exercise 2.37 (p. 50). Let x represent the number of decorated sword shafts buried at a discovered grave site. Explain why x is a discrete random variable.