copyright 2011 brooks/cole, cengage learning random variables class 34 1

Copyright ©2011 Brooks/Cole, Cengage Learning

Random Variables

Class 34

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Copyright ©2011 Brooks/Cole, Cengage Learning 2

8.1 What is a Random Variable?

Random Variable: assigns a number to each outcome of a random circumstance, or, equivalently, to each unit in a population.

Two different broad classes of random variables:1. A discrete random variable can take one of a

countable list of distinct values.2. A continuous random variable can take any

value in an interval or collection of intervals.


Random factors that will determine how enjoyable the event is:

Temperature: continuous random variable

Number of airplanes that fly overhead:discrete random variable

Example 8.1 Random Variables at an Outdoor Graduation or Wedding


8.2 Discrete Random VariablesX the random variable.k = a number the discrete r.v. could assume.P(X = k) is the probability that X equals k.

Probability distribution function (pdf) X is a table or rule that assigns probabilities to possible values of X.

Example: the probability that two girls in the next 3 births at a hospital is 3/8. The random variable X = the number of girls in the next three birthsk = 2 girls (in the next 3 births)P(X = k) = 3/8 (The probability of the number of girls (X) = k (2 girls) in the next 3 births.


8.2 Discrete Random Variables

Example 8.5 Number of Courses35% of students taking four courses, 45% taking five,

and remaining 20% are taking six courses.X = number of courses a randomly selected student is takingThe probability distributionfunction of X can be given by:

One more example of the probability distribution function


Conditions for Probabilities for Discrete Random Variables

Condition 1 The sum of the probabilities over all possible values of a discrete random variable must equal 1.

Condition 2 The probability of any specific outcome for a discrete random variable must be between 0 and 1.


Probability Distribution of a Discrete R.V.Using the sample space to find probabilities:

Step 1: List all simple events in sample space.Step 2: Identify the value of the random variable X

for each simple event.Step 3: Find the probability for each simple event

(often equally likely).Step 4: Find P(X = k) as the sum of the probabilities

for all simple events where X = k.

Probability distribution function (pdf) X is a table or rule that assigns probabilities to possible values of X.


Example 8.6 PDF for Number of Girls Family has 3 children. Probability of a girl is ½.What are the probabilities of having 0, 1, 2, or 3 girls?

Sample Space: For each birth, write either B or G. There are eight possible arrangements of B and G for three births. These are the simple events.

Sample Space and Probabilities: The eight simple events are equally likely.

Random Variable X: number of girls in three births. For each simple event, the value of X is the number of G’s listed.


Example 8.6 & 8.7 Number of Girls

Probability DistributionFunction for Number of Girls X:

Value of X for each simple event:

Graph of the pdf of X:


Cumulative Distribution Function of a Discrete Random Variable

Cumulative distribution function (cdf) for a random variable X is a rule or table that provides the probabilities P(X ≤ k) for any real number k. Cumulative probability = probability that X is less than or equal to a particular value.

Example 8.8 Cumulative Distribution Function for the Number of Girls

Now you try it!

• TRCC offers the freshmen three language courses: Chinese, Spanish and French. What is the probability of a freshman select 0, 1, 2 or 3 language courses? Use a Probability Distribution Function to represent it. Graph both the probability distribution function and the cumulative distribution of it.


Homework• Assignment:• Chapter 8 – Exercise 8.1, 8.9 and 8.11• Reading:• Chapter 8 – p. 265-271

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copyright 2011 brooks/cole, cengage learning random variables class 34 1

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