5.2: Probability Distribution Function (PDF) for a Discrete Random Variable
- Page ID
- 10933
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Introduction
In order to effectively use the formulas in this chapter and the future chapters, we need to emphasize the notion of defining our variables. Specifically, we are going to work with random variables. In this chapter they will be discrete random variables and in the chapter 6 we will look at continous random variables.
Typically, Upper case letters such as \(X\) or \(Y\) denote a random variable. Lower case letters like \(x\) or \(y\) denote the value of a random variable. If \(X\) is a random variable, then \(X\) is written in words, and x is given as a number.
For example, let \(X =\) the number of heads you get when you toss three fair coins.
The sample space for the toss of three fair coins is TTT; THH; HTH; HHT; HTT; THT; TTH; HHH.
Then, \(x =\) 0, 1, 2, 3.
\(X\) is in words and x is a number.
Notice that for this example, the \(x\) values are countable outcomes. Because you can count the possible values that \(X\) can take on and the outcomes are random (the x values 0, 1, 2, 3), \(X\) is a discrete random variable.
However, you may see the lower case and capital letter used interchangeably depending on the textbook and person teaching the course.
Toss a coin ten times and record the number of heads. After all members of the class have completed the experiment (tossed a coin ten times and counted the number of heads), fill in Table. Let \(X =\) the number of heads in ten tosses of the coin.
| \(x\) | Frequency of \(x\) | Relative Frequency of \(x\) |
|---|---|---|
- Which value(s) of \(x\) occurred most frequently?
- If you tossed the coin 1,000 times, what values could \(x\) take on? Which value(s) of \(x\) do you think would occur most frequently?
- What does the relative frequency column sum to?
Discrete Probability Distribution
All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get values such as zero, one, two, or three.
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 | \(P(x = 0) = \dfrac{2}{50}\) |
| 1 | \(P(x = 1) = \dfrac{11}{50}\) |
| 2 | \(P(x = 2) = \dfrac{23}{50}\) |
| 3 | \(P(x = 3) = \dfrac{9}{50}\) |
| 4 | \(P(x = 4) = \dfrac{4}{50}\) |
| 5 | \(P(x = 5) = \dfrac{1}{50}\) |
\(X\) 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,
\[\dfrac{2}{50} + \dfrac{11}{50} + \dfrac{23}{50} + \dfrac{9}{50} + \dfrac{4}{50} + \dfrac{1}{50} = 1\]
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 | \(P(x = 0) = \dfrac{4}{50}\) |
| 1 | \(P(x = 1) = \dfrac{8}{50}\) |
| 2 | \(P(x = 2) = \dfrac{16}{50}\) |
| 3 | \(P(x = 3) = \dfrac{14}{50}\) |
| 4 | \(P(x = 4) = \dfrac{6}{50}\) |
| 5 | \(P(x = 5) = \dfrac{2}{50}\) |
- Answer
-
Each \(P(x)\) is between 0 and 1, inclusive, and the sum of the probabilities is 1, that is:
\[\dfrac{4}{50} + \dfrac{8}{50} +\dfrac{16}{50} +\dfrac{14}{50} +\dfrac{6}{50} + \dfrac{2}{50} = 1\]
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. The table should have two columns labeled \(x\) and \(P(x)\). What does the \(P(x)\) column sum to?
Solutions
a. Let \(X\) = the number of days Nancy attends class per week.
b. x = 0, 1, 2, and 3
c
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.01 |
| 1 | 0.04 |
| 2 | 0.15 |
| 3 | 0.80 |
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 X and what values does it take on?
- Answer
-
\(X\) is the number of days Jeremiah attends basketball practice per week. X takes on the values 0, 1, and 2.
A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have 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.
a. Complete Table using the data provided.
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.12 |
| 1 | 0.18 |
| 2 | 0.30 |
| 3 | 0.15 |
| 4 | |
| 5 | 0.10 |
| 6 | 0.05 |
b. \(P(x \geq 5) =\) _______
c. What does the column “P(x)” sum to?
- Answer
- a.
\(x\) \(P(x)\) 0 0.12 1 0.18 2 0.30 3 0.15 4 0.10 5 0.10 6 0.05 b. \(P(x \geq 5) = 0.10 + 0.05 = 0.15\)
c. \(\Sigma P(X) = 1\)
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 | 0.15 |
| 2 | 0.35 |
| 3 | 0.40 |
| 4 | 0.10 |
a. Define the random variable \(X\).
b. What is the probability the baker will sell exactly one batch? \(P(x = 1) =\) _______
c. What is the probability the baker will sell more than one batch? \(P(x > 1) =\) _______
- Answer
-
a. X = the number of batches of muffins to make to sell in his bakery.
b. \(P(x = 1) = 0.15\)
c. \(P(x > 1) =0.35 + 0.40 + 0.10 = 0.85\)
Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time.
a. Define the random variable \(X\).
b. What values does \(x\) take on?
c. Construct a PDF table.
d. Find the probability that Javier volunteers for less than three events each month. \(P(x < 3) =\) _______
e. Find the probability that Javier volunteers for at least one event each month. \(P(x > 0) =\) _______
Solution
a. Let \(X =\) the number of events Javier volunteers for each month.
b. x = 0, 1, 2, 3, 4, 5 (He could attend any where from 0 to 5 events.)
c.
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.05 |
| 1 | 0.05 |
| 2 | 0.10 |
| 3 | 0.20 |
| 4 | 0.25 |
| 5 | 0.35 |
d. \(P(x < 3) =P(x = 0) + P(x = 1) + P(x=2) = 0.05 + 0.05+0.1 = 0.2\)
e. \(P(x > 0) = 1 - P(x = 0) = 1 – 0.05 = 0.95\) (We can use the complement to save adding up all of the probabilities from \(P(x = 1)\) to \(P(x = 5)\))
Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time. One week is selected at random.
a. Define the random variable \(X\).
b. Construct a probability distribution table for the data.
c. 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?
- Answer
-
a. X = # of days Ellen practice music.
b.
\(x\) \(P(x)\) 0 0.03 1 0.04 2 0.08 3 0.85 c. Each probability is between zero and one, inclusive.
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.
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.