1. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 1 of 15 Chapter 12 Probability and Calculus

2. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 2 of 15  Discrete Random Variables  Continuous Random Variables  Expected Value and Variance  Exponential and Normal Random Variables  Poisson and Geometric Random Variables Chapter Outline

3. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 3 of 15 § 12.1 Discrete Random Variables

4. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 4 of 15  Mean  Expected Value  Variance  Standard Deviation  Frequency Table  Relative Frequency Table  Relative Frequency Histogram  Random Variable  Applications of Expected Value, Variance, and Standard Deviation Section Outline

5. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 5 of 15 Mean DefinitionExample Mean: The sum of a set of numbers, divided by how many numbers were summed

6. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 6 of 15 Expected Value a 1 is the first number from a set of numbers, a 2 is the second and so on p 1 is the probability that a 1 occurs, p 2 is the probability that a 2 occurs and so on

7. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 7 of 15 Variance m is the expected value (or mean) of the set of numbers a 1 is the first number from a set of numbers, a 2 is the second and so on p 1 is the probability that a 1 occurs, p 2 is the probability that a 2 occurs and so on

8. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 8 of 15 Standard Deviation

9. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 9 of 15 Frequency Table (Distribution) DefinitionExample Frequency Table: A list containing a set of numbers and the frequency with which each occurs

10. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 10 of 15 Relative Frequency Table ( Probability Table) DefinitionExample Relative Frequency Table: A list containing a set of numbers and the relative frequency (percent of the time) with which each occurs

11. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 11 of 15 Relative Frequency Histogram DefinitionExample Relative Frequency Histogram: A graph where over each grade we place a rectangle whose height equals the relative frequency of that grade (Compare with relative frequency table from last slide)

12. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 12 of 15 Random Variable DefinitionExample Random Variable: A variable whose value depends entirely on chance Examples will follow.

13. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 13 of 15 Applications of Expected Value, Variance, & Standard DeviationEXAMPLE SOLUTION Find E(X), Var (X), and the standard deviation of X, where X is the random variable whose probability table is given in Table 5.

14. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 14 of 15 Applications of Expected ValueEXAMPLE SOLUTION The number of phone calls coming into a telephone switchboard during each minute was recorded during an entire hour. During 30 of the 1-minute intervals there were no calls, during 20 intervals there was one call, and during 10 intervals there were two calls. A 1-minute interval is to be selected at random and the number of calls noted. Let X be the outcome. Then X is a random variable taking on the values 0, 1, and 2. (a) Write out a probability table for X. (b) Compute E(X). (c) Interpret E(X). Number of Calls012 Probability30/60 = 1/220/60 = 1/310/60 = 1/6 (a)

15. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 15 of 15 Applications of Expected Value (b) (c) Since E(X) = 0.67, this means that the expected number of phone calls in a 1-minute period is 0.67 phone calls. CONTINUED