8 4.1 Introduction to Discrete Random Variables and Notation
Learning Objectives
By the end of this chapter, the student should be able to:
- Recognize and understand discrete probability distribution functions
- Calculate and interpret probabilities, expected values, and standard deviations of general random variables
- Recognize the binomial probability distribution and apply it appropriately
A student takes a ten-question, true-false quiz. Because the student had such a busy schedule, he or she could not study and guesses randomly at each answer. What is the probability of the student passing the test with at least a 70%?
Small companies might be interested in the number of long-distance phone calls their employees make during the peak time of the day. Suppose the average is 20 calls. What is the probability that the employees make more than 20 long-distance phone calls during the peak time?
These two examples illustrate two different types of probability problems involving discrete random variables. Recall that discrete data are data that you can count. A random variable describes the outcomes of a statistical experiment in words. The values of a random variable can vary with each repetition of an experiment.
Random Variables
Random variables are probability models quantifying situations. Upper case letters such as X or Y denote 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 written in words, and x is given as a number.
There are both continuous and discrete random variables. We will begin with discrete RVs and revisit continuous RVs in the future.
Discrete Random Variables
We have seen the word discrete before associated with types of data. Discrete means we have a countable number of outcomes. So a discrete random variable is a RV that models a process or experiment that produces discrete data. Consider the following example of a discrete random variable:
Let X = the number of heads you get when you toss three fair coins. The sample space for the toss of three fair coins is TTT, THH, HTH, HHT, HTT, THT, TTH, HHH. Then, x = 0, 1, 2, 3. X is in words and x is a number. Notice that for this example, the x values are countable outcomes. Because you can count the possible values that X can take on and the outcomes are random (the x values 0, 1, 2, 3), X is a discrete random variable.
Example
A child psychologist is interested in the number of times a newborn baby’s crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. Let X = the number of times per week a newborn baby’s crying wakes its mother after midnight. For this example, x = 0, 1, 2, 3, 4, 5.
P(x) = probability that X takes on a value x.
| x | P(x) |
|---|---|
| 0 | P(x = 0) = [latex]\frac{2}{50}[/latex] |
| 1 | P(x = 1) = [latex]\frac{11}{50}[/latex] |
| 2 | P(x = 2) = [latex]\frac{23}{50}[/latex] |
| 3 | P(x = 3) = [latex]\frac{9}{50}[/latex] |
| 4 | P(x = 4) = [latex]\frac{4}{50}[/latex] |
| 5 | P(x = 5) = [latex]\frac{1}{50}[/latex] |
Your turn!
A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. Let X = the number of times a patient rings the nurse during a 12-hour shift. For this exercise, x = 0, 1, 2, 3, 4, 5. P(x) = the probability that X takes on value x. Is this a discrete probability distribution function (two reasons)?
| X | P(x) |
|---|---|
| 0 | P(x = 0) = [latex]\frac{4}{50}[/latex] |
| 1 | P(x = 1) = [latex]\frac{8}{50}[/latex] |
| 2 | P(x = 2) = [latex]\frac{16}{50}[/latex] |
| 3 | P(x = 3) = [latex]\frac{14}{50}[/latex] |
| 4 | P(x = 4) = [latex]\frac{6}{50}[/latex] |
| 5 | P(x = 5) = [latex]\frac{2}{50}[/latex] |
Characteristics and Notation
The distribution of a discrete random variable is often pictured in a table, but may also be represented by a graph or formula. Two main characteristics it should exhibit are:
- Each probability is between zero and one, inclusive.
- The sum of the probabilities is one.
The probability mass function (PMF) of a DRV tells you the probability of a single value. Notation-wise this means P(X = x). This is also sometimes (erroneously) called probability distribution function (PDF).
The cumulative distribution function (CDF) of a DRV tells you the probability of being less than or equal to a value. Notation-wise this means P(X ≤ x).
A probability distribution function is a pattern. You try to fit a probability problem into a pattern or distribution in order to perform the necessary calculations. These distributions are tools to make solving probability problems easier. Each distribution has its own special characteristics. Learning the characteristics enables you to distinguish among the different distributions.
Example
Suppose Nancy has classes three days a week. She attends classes three days a week 80% of the time, two days 15% of the time, one day 4% of the time, and no days 1% of the time. Suppose one week is randomly selected.
a. Let X = the number of days Nancy .
b. X takes on what values?
c. Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table) like the one below. The table should have two columns labeled x and P(x). What does the P(x) column sum to?
| x | P(x) |
|---|---|
| 0 | |
| 1 | |
| 2 | |
| 3 |
d. Construct the cumulative probability distribution function
Your turn!
Jeremiah has basketball practice two days a week. Ninety percent of the time, he attends both practices. Eight percent of the time, he attends one practice. Two percent of the time, he does not attend either practice. What is X and what values does it take on?
Image Credits
Figure 4.1: Michael D (2018). “Storm at dawn.” Public domain. Retrieved from https://unsplash.com/photos/2cDIzRnVq0Q
A representation of a probability model
A mathematical representation of a random process that lists all possible outcomes and assigns probabilities to each of them
A random variable that produces discrete data
Data produced by a variable that takes on an uncountable, infinite, number of values
A function that gives the probability that a discrete random variable is exactly equal to some value (x)
A function that gives the probability that a random variable takes a value less than or equal to x
