Discrete Random Variables
18 Random Variable – Discrete
A discrete probability distribution function has two characteristics:
- Each probability is between zero and one, inclusive.
- The sum of the probabilities is one.
Example 4.1
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 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 | $P(x=0) = \frac{2}{50}$ |
| 1 | $P(x=1) = \frac{11}{50}$ |
| 2 | $P(x=2) = \frac{23}{50}$ |
| 3 | $P(x=3) = \frac{9}{50}$ |
| 4 | $P(x=4) = \frac{4}{50}$ |
| 5 | $P(x=5) = \frac{1}{50}$ |
X takes on the values 0, 1, 2, 3, 4, 5. This is a discrete PDF because:
- Each P(x) is between zero and one, inclusive.
- The sum of the probabilities is one, that is,
$\frac{2}{50}+\frac{11}{50}+\frac{23}{50}+\frac{9}{50}+\frac{4}{50}+\frac{1}{50}=1$
Try It 4.1
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. Why is this a discrete probability distribution function (two reasons)?
| X | P(x) |
|---|---|
| 0 | $P(x=0) = \frac{4}{50}$ |
| 1 | $P(x=1) = \frac{8}{50}$ |
| 2 | $P(x=2) = \frac{16}{50}$ |
| 3 | $P(x=3) = \frac{14}{50}$ |
| 4 | $P(x=4) = \frac{6}{50}$ |
| 5 | $P(x=5) = \frac{2}{50}$ |
Example 4.2
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.
- Let X = the number of days Nancy ____________________.
- X takes on what values?
- Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table) like the one in Example 4.1. The table should have two columns labeled x and P(x). What does the P(x) column sum to?
