Connexions module: m16831
1
Discrete Random Variables:
Probability Distribution Function
(PDF) for a Discrete Random
Variable
∗
Susan Dean
Barbara Illowsky, Ph.D.
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License
†
Abstract
This module introduces the Probability Distribution Function (PDF) and its characteristics.
A discrete
•
•
probability distribution function has two characteristics:
Each probability is between 0 and 1, inclusive.
The sum of the probabilities is 1.
P(X) is the notation used to represent a discrete
probability distribution function.
Example 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 a newborn wakes its mother after midnight. For this example,
x
= 0, 1,
2, 3, 4, 5.
P(X = x) = probability that
∗ Version
X
takes on a value
x.
x
P(X = x)
0
P(X=0)
=
1
P(X=1)
=
2
P(X=2)
=
3
P(X=3)
=
4
P(X=4)
=
5
P(X=5)
=
2
50
11
50
23
50
9
50
4
50
1
50
1.11: Jan 25, 2009 5:59 pm US/Central
† http://creativecommons.org/licenses/by/2.0/
Source URL: http://cnx.org/content/col10522/latest/
Saylor URL: http://www.saylor.org/courses/ma121/
http://cnx.org/content/m16831/1.11/
Attributed to: Barbara Illowsky and Susan Dean
Saylor.org
Page 1 of 3
Connexions module: m16831
2
Table 1
X
takes on the values 0, 1, 2, 3, 4, 5. This is a discrete
PDF
because
1. Each P(X = x) is between 0 and 1, inclusive.
2. The sum of the probabilities is 1, that is,
2
11 23
9
4
1
+
+
+
+
+
=1
50 50 50 50 50 50
(1)
Example 2
Suppose Nancy has classes 3 days a week. She attends classes 3 days a week 80% of the time, 2
days 15% of the time, 1 day 4% of the time, and no days 1% of the time.
Problem 1
(Solution on p. 3.)
Let
X
= the number of days Nancy ____________________ .
Problem 2
X
(Solution on p. 3.)
takes on what values?
Problem 3
(Solution on p. 3.)
Construct a probability distribution table (called a
The table should have two columns labeled
PDF table) like the one in the previous example.
x and P(X
= x). What does the P(X = x) column sum
to?
Source URL: http://cnx.org/content/col10522/latest/
Saylor URL: http://www.saylor.org/courses/ma121/
http://cnx.org/content/m16831/1.11/
Attributed to: Barbara Illowsky and Susan Dean
Saylor.org
Page 2 of 3
Connexions module: m16831
3
Solutions to Exercises in this Module
Solution to Example 2, Problem 1 (p. 2)
Let X = the number of days Nancy attends class per week.
Solution to Example 2, Problem 2 (p. 2)
0, 1, 2, and 3
Solution to Example 2, Problem 3 (p. 2)
x
P (X = x)
0
0.01
1
0.04
2
0.15
3
0.80
Table 2
Glossary
Denition 1: 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.
Example
A biased coin with probability 0.7 for a head (in one toss of the coin) is tossed 5 times.
are interested in the number of heads (the RV

X ∼ B (5, 0.7)
and
P (X = x) =
5
x
X
= the number of heads).
X
We
is Binomial, so

 .7x .35−x or
in the form of the table:
x
P (X = x)
0
0.0024
1
0.0284
2
0.1323
3
0.3087
4
0.3602
5
0.1681
Table 3
Source URL: http://cnx.org/content/col10522/latest/
Saylor URL: http://www.saylor.org/courses/ma121/
http://cnx.org/content/m16831/1.11/
Attributed to: Barbara Illowsky and Susan Dean
Saylor.org
Page 3 of 3