Module 6: Probability and Probability Distributions
Module 6: Probability and Probability Distributions
Discrete Random Variables (5 of 5)
Discrete Random Variables (5 of 5)
Learning OUTCOMES
- Use probability distributions for discrete and continuous random variables to estimate probabilities and identify unusual events.
Here is another example of how to use the mean and standard deviation of a discrete random variable to identify unusual values for a random variable.
Example
Changing Majors
Here we have again the probability distribution of the number of changes in major.
| X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| P(X) | 0.135 | 0.271 | 0.271 | 0.180 | 0.090 | 0.036 | 0.012 | 0.003 | 0.002 |
How often do we expect a college student to change majors?
This question is asking for the expected value, which is the mean of the probability distribution. So we calculate the weighted average, as before:
[latex]0(0.135) + 1(0.271) + 2(0.271) + 3(0.180) + 4(0.090) + 5(0.036) + 6(0.012) + 7(0.003) + 8(0.002) = 2[/latex]
What is the standard deviation of the probability distribution?
[latex]\sqrt{(0-2)^2 (0.135) + (1-2)^2 (0.271) + (2 - 2)^2 (0.271) + (3-2)^2 (0.180) + (4-2)^2 (0.090) + \text{...} + (7-2)^2 (0.003) + (8-2)^2 (0.002)} \approx 1.4[/latex]
We have drawn lines to show the mean and 1 standard deviation above and below the mean.
Recall that earlier, we discussed what would be considered an unusual (and not unusual) number of changes in major, and we used probability calculations to assess that. For example, we found that changing majors 5 or more times occurs only about 5% of the time and therefore can be considered unusual.
Another way to think about defining “unusual” is to look at outcomes relative to the mean. We might consider outcomes more than 2 standard deviations above the mean as unusual.
What values are more than 2 standard deviations above the mean of 2?
Mean + 2 (standard deviation) = [latex]\mu_x + 2 \cdot \text{SD} = 2 + 2 \cdot 1.4 = 4.8[/latex], which rounds to 5.
We conclude from this line of reasoning that a college student who changes majors 5 or more times is “unusual.”
In Summarizing Data Graphically and Numerically, we used the standard deviation to identify usual, or typical, values. We said that a typical range of values falls within 1 standard deviation of the mean. We can use a similar idea here.
Try It
Example
Detecting Fraud
Legitimate records often display a surprising pattern that is not present in faked tax returns or other fraudulent accounting records. In legitimate records, the distribution of first digits can be modeled using Benford’s law. For example, suppose the total income recorded on a tax return is $20,712. The first digit is 2. Now we examine a very large number of tax returns and record the first digit of total income for all of the returns. The relative frequency of each first digit will behave according to Benford’s law.
Benford’s law can also be described using a mathematical formula, but we will not go into that here. Instead, let’s double-check th