Probability Distribution Function (PDF) for a Discrete Random Variable, Discrete Random Variables | Statistics

Learning objectives

There are two types of random variables, discrete random variables and continuous random variables. The values of a discrete random variable are countable, which means the values are obtained by counting. All random variables we discussed in previous examples are discrete random variables. We counted the number of red balls, the number of heads, or the number of female children to get the corresponding random variable values. The values of a continuous random variable are uncountable, which means the values are not obtained by counting. Instead, they are obtained by measuring. For example, let X = temperature of a randomly selected day in June in a city. The value of X can be 68°, 71.5°, 80.6°, or 90.32°. These values are obtained by measuring by a thermometer. Another example of a continuous random variable is the height of a randomly selected high school student. The value of this random variable can be 5'2", 6'1", or 5'8". Those values are obtained by measuring by a ruler.

A discrete probability distribution function has two characteristics:

Each probability is between zero and one, inclusive.
The sum of the probabilities is one.

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 we can count the number of values of x and also because of the following two reasons:

Each P(x) is between zero and one, therefore 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

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) = $\frac{2}{50}$

Each P(x) is between 0 and 1, inclusive, and the sum of the probabilities is 1, that is: $\frac{4}{50} + \frac{8}{50} + \frac{16}{50} + \frac{14}{50} + \frac{6}{50} + \frac{2}{50} = 1$

Suppose Nancy has classes three days a week. She attends classes three days a week 80 percent of the time, two days 15 percent of the time, one day 4 percent of the time, and no days 1 percent of the time. Suppose one week is randomly selected.

Describe the random variable in words. Let X = the number of days Nancy ________.

a. Let X = the number of days Nancy attends class per week.

b. In this example, what are possible values of X?

b. 0, 1, 2, and 3

c. Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table) like the one in example1. The table should have two columns labeled x and P(x).

x	P(x)
0	.01
1	.04
2	.15
3	.80

The sum of the P(x) column is 0.01+0.04+0.15+0.80 = 1.00.

Chapter Review

The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows:

Each probability is between zero and one, inclusive (inclusive means to include zero and one)
The sum of the probabilities is one

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, or in other words, how long new hires stay with the company. Over the years, the company has established the following probability distribution:

Let X = the number of years a new hire will stay with the company.

Let P(x) = the probability that a new hire will stay with the company x years.

Complete M02_Ch03_tbl004 using the data provided.

x	P(x)
0	.12
1	.18
2	.30
3	.15
4
5	.10
6	.05

x	P(x)
0	.12
1	.18
2	.30
3	.15
4	.10
5	.10
6	.05

P(x = 4) = ________

P(x ≥ 5) = ________

.10 + .05 = .15

On average, how long would you expect a new hire to stay with the company?

What does the column “P(x)” sum to?

Use the following information to answer the next four exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution.

x	P(x)
1	.15
2	.35
3	.40
4	.10

Define the random variable X.

What is the probability the baker will sell more than one batch? P(x > 1) = ________

.35 + .40 + .10 = .85

What is the probability the baker will sell exactly one batch? P(x = 1) = ________

On average, how many batches should the baker make?

1(.15) + 2(.35) + 3(.40) + 4(.10) = .15 + .70 + 1.20 + .40 = 2.45

Use the following information to answer the next two exercises: Ellen has music practice three days a week. She practices for all of the three days 85 percent of the time, two days 8 percent of the time, one day 4 percent of the time, and no days 3 percent of the time. One week is selected at random.

Define the random variable X.

Construct a probability distribution table for the data.

x	P(x)
0	.03
1	.04
2	.08
3	.85

We know that for a probability distribution function to be discrete, it must have two characteristics. One is that the sum of the probabilities is one. What is the other characteristic?

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35 percent of the time, four events 25 percent of the time, three events 20 percent of the time, two events 10 percent of the time, one event 5 percent of the time, and no events 5 percent of the time.

Define the random variable X.

Let X = the number of events Javier volunteers for each month.

What values does x take on?

Construct a PDF table.

x	P(x)
0	.05
1	.05
2	.10
3	.20
4	.25
5	.35

Find the probability that Javier volunteers for fewer than three events each month. P(x < 3) = ________

Find the probability that Javier volunteers for at least one event each month. P(x > 0) = ________

1 – .05 = .95

Homework

Suppose that the PDF for the number of years it takes to earn a bachelor of science (B.S.) degree is given in M03_Ch04_tbl010.

x	P(x)
3	.05
4	.40
5	.30
6	.15
7	.10

In words, define the random variable X.
What does it mean that the values 0, 1, and 2 are not included for x in the PDF?

Glossary

probability distribution function (PDF): a mathematical description of a discrete random variable (RV), given either in the form of an equation (formula) or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome

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