5.2: The Probability Distribution Function
- Page ID
- 26058
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A discrete probability distribution function has two characteristics:
- Each probability is between zero and one, inclusive.
- The sum of the probabilities is one.
Example \(\PageIndex{1}\)
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 | \(P(x = 0) = \dfrac{2}{50}\) |
| 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,
\[\dfrac{2}{50} + \dfrac{11}{50} + \dfrac{23}{50} + \dfrac{9}{50} + \dfrac{4}{50} + \dfrac{1}{50} = 1\]
Example \(\PageIndex{2}\)
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. 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. 0, 1, 2, and 3
c
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.01 |
| 1 | 0.04 |
| 2 | 0.15 |
| 3 | 0.80 |
WeBWorK Problems
Query \(\PageIndex{1}\)
Query \(\PageIndex{2}\)
Query \(\PageIndex{3}\)
Query \(\PageIndex{4}\)
Query \(\PageIndex{5}\)
Query \(\PageIndex{6}\)
Query \(\PageIndex{7}\)
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.
Contributors and Attributions
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/30189442-699...b91b9de@18.114.
Use the following information to answer the next five exercises: 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.
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.