1. Copyright ©2011 Brooks/Cole, Cengage Learning Random Variables Class 34 1

2. 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.

3. Copyright ©2011 Brooks/Cole, Cengage Learning 3 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

4. Copyright ©2011 Brooks/Cole, Cengage Learning 4 8.2 Discrete Random Variables X 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 births k = 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.

5. Copyright ©2011 Brooks/Cole, Cengage Learning 5 8.2 Discrete Random Variables Example 8.5 Number of Courses 35% of students taking four courses, 45% taking five, and remaining 20% are taking six courses. X = number of courses a randomly selected student is taking The probability distribution function of X can be given by: One more example of the probability distribution function

6. Copyright ©2011 Brooks/Cole, Cengage Learning 6 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.

7. Copyright ©2011 Brooks/Cole, Cengage Learning 7 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.

8. Copyright ©2011 Brooks/Cole, Cengage Learning 8 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.

9. Copyright ©2011 Brooks/Cole, Cengage Learning 9 Example 8.6 & 8.7 Number of Girls Probability Distribution Function for Number of Girls X: Value of X for each simple event: Graph of the pdf of X:

10. Copyright ©2011 Brooks/Cole, Cengage Learning 10 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

11. 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. Copyright ©2011 Brooks/Cole, Cengage Learning 11

12. Homework Assignment: Chapter 8 – Exercise 8.1, 8.9 and 8.11 Reading: Chapter 8 – p. 265-271 12