Discrete Random Variables: Probability Distribution Function (PDF) for ... [PDF]

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

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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

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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