Chapter 4: Discrete Random Variables
4.2 Measures of General Discrete Random Variables
Learning Objectives
By the end of this section, the student should be able to:
- Calculate and interpret expected values of general random variables
- Calculate and interpret the variance and standard deviation of general random variables
Once we know how to work with Discrete Random Variables we may be interested in some other measures such as the mean, variance, and standard deviation. The ideas here are slightly different than we have seen before within our new context of Random Variables.
The Expected Value (Mean) of a Discrete Random Variable
The Law of Large Numbers states: as the number of trials in a probability experiment increases, our results become closer to what we “expect.” For instance, the student who was guessing on the true-false quiz in the chapter introduction would expect to get about half of the questions correct, since there are two options.
When evaluating the long-term results of statistical experiments, we often want to know the “average” outcome. This long-term average is known as the mean or expected value of the random variable and is denoted by the Greek letter [latex]\mu[/latex], or in the context of random variables, [latex]E[X][/latex]. In other words, after conducting many trials of an experiment, you would expect this average value.
To find the expected value, we multiply each value of the random variable by its probability, then add the products.
Mean or Expected Value: [latex]\mu={\sum\limits_{x \in X} }^{\text{}}xP\left(x\right).[/latex]
Example
A university soccer team plays soccer zero, one, or two days a week. The probability that they play zero days is 0.2, the probability that they play one day is 0.5, and the probability that they play two days is 0.3. Find the long-term average or expected value, [latex]\mu[/latex], of the number of days per week the team plays soccer.
We first let the random variable [latex]X[/latex] = the number of days the team plays soccer per week. Then [latex]x[/latex] takes on the values 0, 1, 2. Construct a PDF table adding a column [latex]x \cdot P(x)[/latex]. In this column, you multiply each value by its probability.
| [latex]x[/latex] | [latex]P(x)[/latex] | [latex]x \cdot P(x)[/latex] |
|---|---|---|
| 0 | 0.2 | (0)(0.2) = 0 |
| 1 | 0.5 | (1)(0.5) = 0.5 |
| 2 | 0.3 | (2)(0.3) = 0.6 |
What is the expected value?
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. What is the expected value?
| [latex]x[/latex] | [latex]P(x)[/latex] |
|---|---|
| 0 | [latex]P(0)=\frac{4}{50}[/latex] |
| 1 | [latex]P(1) = \frac{8}{50}[/latex] |
| 2 | [latex]P(2) = \frac{16}{50}[/latex] |
| 3 | [latex]P(3) = \frac{14}{50}[/latex] |
| 4 | [latex]P(4) = \frac{6}{50}[/latex] |
| 5 | [latex]P(5) = \frac{2}{50}[/latex] |
The Variance and Standard Devi