Chapter 4: Discrete Random Variables
4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
Learning Objectives
By the end of this section, you should be able to:
- Validate the properties of a discrete probability distribution function
- Write a probability distribution function for given discrete data
Random Variables
Random variables are probability models quantifying situations. Upper case letters such as [latex]X[/latex] or [latex]Y[/latex] denote a random variable. Lower case letters like [latex]x[/latex] or [latex]y[/latex] denote the value of a random variable. If [latex]X[/latex] is a random variable, then [latex]X[/latex] is written in words, and [latex]x[/latex] is given as a number.
For example, let [latex]X[/latex] = 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, in these outcomes, we can have no heads (when TTT occurs) or up to three heads (when HHH occurs). Therefore [latex]x = 0, 1, 2,[/latex] and [latex]3[/latex].
Observe again that [latex]X[/latex] is in words and [latex]x[/latex] is a number. For this example, the [latex]x[/latex] values are a countable number of outcomes, having four possibilities. Because you can count the possible values that [latex]X[/latex] can take on and the outcomes are random, [latex]X[/latex] is a discrete random variable.
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 random variable that models a process or experiment that produces discrete data.
There are both continuous and discrete random variables. We will begin with discrete random variables and examine continuous random variables in the next chapter.
Characteristics and Notation
The probability mass function (PMF) of a discrete random variable tells you the probability of each outcome in the sample space. The probability that random variable [latex]X[/latex] takes on value [latex]x[/latex] is represented by [latex]P(X = x)[/latex] or just [latex]P(x)[/latex]. The PMF is also sometimes called the probability distribution function (PDF).
The distribution of a discrete random variable is often pictured in a table, but may also be represented by a graph or formula. The two main characteristics the PMF should exhibit are:
- Each probability is between zero and one, inclusive.
- The sum of the probabilities is one.
Example
A child psychologist is interested in the number of times a newborn baby’s crying wakes one of their parents after midnight. For a random sample of 50 families, the following information was obtained. Let X = the number of times per week a newborn baby’s crying wakes one of their parents after midnight. In the sample, the largest number of times observed during a week was 5, so x = 0, 1, 2, 3, 4, 5.
P(x) = probability that X takes on a value x.
| x | P(x) |
|---|---|
| 0 | P(0) = [latex]\frac{2}{50}[/latex] |
| 1 | P(1) = [latex]\frac{11}{50}[/latex] |
| 2 | P(2) = [latex]\frac{23}{50}[/latex] |
| 3 | P(3) = [latex]\frac{9}{50}[/latex] |
| 4 | P(4) = [latex]\frac{4}{50}[/latex] |
| 5 | P(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. The largest number of patient calls was 5, so x = 0, 1, 2, 3, 4, 5.
| X | P(x) |
|---|---|
| 0 | P(0) = [latex]\frac{4}{50}[/latex] |
| 1 | P(1) = [latex]\frac{8}{50}[/latex] |
| 2 | P(2) = [latex]\frac{16}{50}[/latex] |
| 3 | P(3) = [latex]\frac{14}{50}[/latex] |
| 4 | P(4) = [latex]\frac{6}{50}[/latex] |
| 5 | P(5) = [latex]\frac{2}{50}[/latex] |
The cumulative distribution function (CDF) of a discrete random variable tells you the probability of being less than or equal to a value, denoted [latex]P(X \leq x)[/latex].
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?
A probability model quantifying a situation; can equal the values of the possible outcomes. Represented using a capital letter.
A mathematical representation of a random process that lists all possible outcomes and assigns probabilities to each of them
a random variable (RV) whose outcomes are counted
A random variable with a countable number of outcomes
A representation of a probability model
A random variable (RV) whose outcomes are measured as an uncountable, infinite, number of values
A function that gives the probability that a discrete random variable (X) 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