7.1 Probability Distribution of a Discrete Random Variable
Learning Objectives
- Recognize, understand, and construct discrete probability distributions.
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.
Video: “Random Variables and Probability Distributions” by Dr Nic’s Maths and Stats [4:39] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.
The probability distribution for a random variable lists all the possible values of the random variable and the probability the random variable takes on each value. The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. A probability distribution can be a table, with a column for the values of the random variable and another column for the corresponding probability, or a graph, like a histogram with the values of the random variable on the horizontal axis and the probabilities on the vertical axis.
In a probability distribution, each probability is between 0 and 1, inclusive. Because all possible values of the random variable are included in the probability distribution, the sum of the probabilities is 1.
Example 7.1.1
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 [latex]X[/latex] be the number of times per week a newborn baby’s crying wakes its mother after midnight. For this example, the values of the random variable are [latex]x = 0, 1, 2, 3, 4, 5[/latex].
In the table, the left column contains all of the possible values of the random variable and the right column, [latex]P(x)[/latex], is the probability that [latex]X[/latex] takes on the corresponding value [latex]x[/latex]. For example, in the first row, the value of the random variable is 0, and the probability the random variable is 0 is [latex]\displaystyle{\frac{2}{50}}[/latex]. In the context of this example, that means that the probability a newborn baby’s crying wakes its mother 0 times per week is [latex]\displaystyle{\frac{2}{50}}[/latex].
| [latex]x[/latex] | [latex]P(x)[/latex] |
| 0 | [latex]\frac{2}{50}[/latex] |
| 1 | [latex]\frac{11}{50}[/latex] |
| 2 | [latex]\frac{23}{50}[/latex] |
| 3 | [latex]\frac{9}{50}[/latex] |
| 4 | [latex]\frac{4}{50}[/latex] |
| 5 | [latex]\frac{1}{50}[/latex] |
Because [latex]X[/latex] can only take on the values 0, 1, 2, 3, 4, and 5, [latex]X[/latex] is a discrete random variable. Note that each probability is between 0 and 1 and the sum of the probabilities is 1:
Try It
Suppose Nancy has classes three days a week. She attends classes three days a week 80% of the time, two days a week 15% of the time, one day a week 4% of the time, and no days 1% of the time. Suppose one week is randomly selected.
- Let [latex]X[/latex] be the number of days Nancy ____________________.
- [latex]X[/latex] takes on what values?
- Suppose one week is randomly chosen. Construct a probability distribution table like the one in the example above. The table should have two columns labelled [latex]x[/latex] and [latex]P(x)[/latex]. What does the [latex]P(x)[/latex] column sum to?
Solution
- Let [latex]X[/latex] be the number of days Nancy attends class per week.
- 0, 1, 2, and 3.
-
[latex]x[/latex] [latex]P(x)[/latex] 0 0.01 1 0.04 2 0.15 3 0.80 The [latex]P(x)[/latex] column sums to 1.
Example 7.1.2
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 [latex]X[/latex], and what values does it take on?
Solution:
[latex]X[/latex] is the number of days Jeremiah attends basketball practice per week. [latex]X[/latex] takes on the values 0, 1, and 2.
Video: “Constructing a probability distribution for random variable | Khan Academy” by Khan Academy [6:47] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.
Key Takeaways
Attribution
“4.2 Probability Distribution of a Discrete Random Variable” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
