1. 1-1 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 1 Chapter 16 Random Variables

2. 1-2 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 2 Introductory Thoughts Statistics holds the key to MANY of life’s key questions, such as the following: How do insurance companies know how much to charge people? If you are putting on a raffle, how much do you charge per ticket to make sure that you make money? If you roll four dice, what is the most likely sum you will receive?

3. 1-3 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 3 Expected Value: Center A random variable assumes a value based on the outcome of a random event. We use a capital letter, like X, to denote a random variable. A particular value of a random variable will be denoted with the corresponding lower case letter, in this case x.

4. 1-4 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 4 Expected Value: Center (cont.) There are two types of random variables: Discrete random variables can take one of a countable number of distinct outcomes. You could sit down and list the possible outcomes. Example: number of credit hours Continuous random variables can take any numeric value within a range of values. You couldn’t sit and list all the possible outcomes. Example: age of college seniors

5. 1-5 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 5 Example: Insurance Payouts Policyholder OutcomePayout (x)Probability: P( X = x) Death10,0001/1000 Disability5,0002/1000 Neither0997/1000 What is the random variable? Is this variable discrete or continuous? Click to the next slide, come back and ask a particular question that hopefully you remember.

6. 1-6 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 6 Expected Value: Center (cont.) A probability model for a random variable consists of: The collection of all possible values of a random variable, and the probabilities that the values occur. Of particular interest is the value we expect a random variable to take on, notated μ (for population mean) or E(X) for expected value. NOTE: We are using a model here, not an actual sample of data. Why can’t we use the same formulas as before or x ???????????

7. 1-7 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 7 Expected Value: Center (cont.) The expected value of a discrete random variable can be found by summing the products of each possible value by the probability that it occurs: Note: Be sure that every possible outcome is included in the sum and verify that you have a valid probability model to start with.

8. 1-8 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 8 Example: Insurance Payouts Let’s calculate the expected value of an insurance payout with this company’s information: Policyholder OutcomePayout (x)Probability: P( X = x) Death10,0001/1000 Disability5,0002/1000 Neither0997/1000 We should get E(X) = $20. Is the company losing or gaining money with these rates?

9. 1-9 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 9 Just Checking – Page 368 Break up into groups and work this problem.

10. 1-10 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 10 First Center, Now Spread… For data, we calculated the standard deviation by first computing the deviation from the mean and squaring it. We do that with discrete random variables as well. The variance for a random variable is: The standard deviation for a random variable is:

11. 1-11 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 11 Example: Insurance Payouts Policyholder Outcome Payout (x) P( X = x)Deviation x – E(X) Death10,0001/10009980 Disability5,0002/10004980 Neither0997/1000-20 We should get Var(X) = 149600, SD(X) = 386.78

12. 1-12 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 12 AP EXAM TIP Both of these formulas are on the official AP Statistics formula sheet, which is allowed on the exam. We will become accustomed to answering this question on the calculator, however. Page 371.

13. 1-13 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 13 Teaching Tip While expected value is easy enough to find “by hand”, finding variance and standard deviation is probably best done using the frequency list feature of a calculator. Students can write a term or two of the formula to communicate their understanding of the procedure. Both of these formulas are on the official AP Statistics formula sheet, which is allowed on the exam.

14. 1-14 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 14 More About Means and Variances Adding or subtracting a constant from data shifts the mean but doesn’t change the variance or standard deviation: E(X ± c) = E(X) ± c Var(X ± c) = Var(X) This is the same as before when we did this. $50 premium for previous problem Consider everyone in a company receiving a $5000 increase in salary

15. 1-15 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 15 More About Means and Variances (cont.) In general, multiplying each value of a random variable by a positive constant multiplies the mean by that constant and the variance by the square of the constant: E(aX) = aE(X)Var(aX) = a 2 Var(X) SD(aX) = aSD(X) Example: Consider everyone in a company receiving a 10% increase in salary.

16. 1-16 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 16 Insurance Example We found the average payout (or expected value of a payout) for a random policyholder to be $20. If you looked at two different customers at the same time, what would you suppose to be the companies expected payout for both of these individuals combined? What if you subtracted one’s payout from another?

17. 1-17 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 17 More About Means and Variances (cont.) In general, The mean of the sum of two random variables is the sum of the means. The mean of the difference of two random variables is the difference of the means. E(X ± Y) = E(X) ± E(Y)

18. 1-18 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 18 Insurance Example Would we expect variance to be the same? Let’s think about the same two policyholders. Is the outcome of their payouts independent of each other? Is it likely that they will both die in the same year? What is the spread of the possible outcomes? The fact is that if one person dies, it is still very unlikely the other person dies. We are able to just add their variances: VAR( X + Y) = VAR(X) + VAR(Y) = 149600 + 149600 = 299200. CALCULATION NOTE!

19. 1-19 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 19 Insurance Example What if we just double the coverage for one policyholder? Is the variance of this, VAR(2X), the same thing as VAR(X+Y)? If not, which is better for the insurance company?  One more situation: What if we take the variance of one policyholder minus another? VAR(X-Y)?  What is the range of the outcomes?

20. 1-20 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 20 More About Means and Variances (cont.) In general, If the random variables are independent, the variance of their sum or difference is always the sum of the variances. Var(X ± Y) = Var(X) + Var(Y) My research tells me that adding variances has been the key to several free response questions on the AP exam in the past, so we mustn’t ignore it.

21. 1-21 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 21 Need to Know E(X ± c) = E(X) ± c Var(X ± c) = Var(X) E(aX) = aE(X) Var(aX) = a 2 Var(X) SD(aX) = aSD(X) E(X ± Y) = E(X) ± E(Y) Var(X ± Y) = Var(X) + Var(Y)

22. 1-22 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 22 Food and Love On Valentine’s Day, a restaurant offers a discount to couples: with your check, the waiter brings four aces from a deck of cards. The couple can turn over one card. If this card is a black ace, the gentleman owes the full amount. If it is the ace of hearts, he gets a discount of $20. If it is the ace of diamonds, they can draw again (w/out replacement) and if they get the ace of hearts this time, he gets a $10 discount. Construct a probability model for this problem, and find E(X), VAR(X), and SD(X). If they offer a $5 discount on top of these savings (1 per table), what would the expected value and the standard deviation of the discounts be then?

23. 1-23 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 23 Food and Love Suppose two couples dine together on one check. The restaurant states they will double the original discounts: $40 for getting the ace of hearts the first time, $20 for the 2 nd. What is the mean and the standard deviation now? What if these two couples decided to pay separately instead of together? Would there be any difference in the outcome? If so, which one would be the smarter choice for the couples?

24. 1-24 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 24 Food and Love Suppose the restaurant down the road offers a competitive deal they claim to be better. Its discounts have a mean of $10 with a standard deviation of $15. How much more would you expect to save at this restaurant than the first one? With what standard deviation? The owner of the first restaurant is planning on serving 40 couples (with the original savings). What is the expected total discounts to be given? With what standard deviation?

25. 1-25 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 25 Looking ahead In Chapters 21 and 23 we will use these variance formulas to help us study the difference between two proportions and two means. The variance formula should feel a bit familiar. Finding a new standard deviation works just like the Pythagorean Theorem….

26. 1-26 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 26 Continuous Random Variables Random variables that can take on any value in a range of values are called continuous random variables. The Normal Model is a continuous random variable.

27. 1-27 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 27 Continuous Random Variables (cont.) Good news: nearly everything we’ve said about how discrete random variables behave is true of continuous random variables, as well. When two independent continuous random variables have Normal models, so does their sum or difference. This fact will let us apply our knowledge of Normal probabilities to questions about the sum or difference of independent random variables.

28. 1-28 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 28 Normal Distribution on the Calculator If the data is normal and standardized, find the probability that… Data value is between -1.63 and 0.73 z-scores. Data value is less than 2.01 z-score. Data value is more than -1.21 z-score.

29. 1-29 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 29 What Can Go Wrong? Probability models are still just models. Models can be useful, but they are not reality. Question probabilities as you would data, and think about the assumptions behind your models. If the model is wrong, so is everything else.

30. 1-30 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 30 What Can Go Wrong? (cont.) Don’t assume everything’s Normal. You must Think about whether the Normality Assumption is justified. Watch out for variables that aren’t independent: You can add expected values for any two random variables, but you can only add variances of independent random variables.

31. 1-31 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 31 What Can Go Wrong? (cont.) Don’t forget: Variances of independent random variables add. Standard deviations don’t. Don’t forget: Variances of independent random variables add, even when you’re looking at the difference between them. Don’t write independent instances of a random variable with notation that looks like they are the same variables.

32. 1-32 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 32 What have we learned? We know how to work with random variables. We can use a probability model for a discrete random variable to find its expected value and standard deviation. The mean of the sum or difference of two random variables, discrete or continuous, is just the sum or difference of their means. And, for independent random variables, the variance of their sum or difference is always the sum of their variances.

33. 1-33 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 33 What have we learned? (cont.) Sums or differences of Normally distributed random variables also follow Normal models.

34. 1-34 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 15, Slide 34 AP Tips Finding expected value from a probability distribution is a common multiple choice question on the AP exam. If you are creating a probability model, make sure your probabilities sum to 1. Make sure to think about independence before you find the standard deviation of the combination of two random variables.