2. Copyright © 2010 Pearson Education, Inc. Chapter 16 Random Variables

3. Copyright © 2010 Pearson Education, Inc. Greedy Pig Everyone stand Everyone gets the points on the dice roll. At each turn you can sit down and keep your points, or remain standing and add to your score. If a 5 is rolled, everyone standing loses all their points. Play 5 rounds – who gets the highest total score? There is an optimal strategy! Slide 16 - 3

4. Copyright © 2010 Pearson Education, Inc. Would you play… A player pays $5 to play the game. Using a regular deck of cards the player draws one card at random Ace of hearts wins $100 Any other Ace wins $10 Any other heart wins $5 Anything else loses. What if the top prize was $200? Slide 16 - 4

5. Copyright © 2010 Pearson Education, Inc. Slide 16 - 5 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.

6. Copyright © 2010 Pearson Education, Inc. Slide 16 - 6 Expected Value: Center (cont.) There are two types of random variables: Discrete random variables can take one of a countable number of distinct outcomes. Example: Number of credit hours Continuous random variables can take any numeric value within a range of values. Example: Cost of books this term

7. Copyright © 2010 Pearson Education, Inc. Slide 16 - 7 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.

8. Copyright © 2010 Pearson Education, Inc. Slide 16 - 8 Expected Value: Center (cont.) An insurance company offers a policy that pays $10,000 if you die and $5,000 if you are permanently disabled. The policy costs $50 per year. Will the company make a profit? Suppose the death rate is 1 per 1,000, and the disability rate is 2 per 1,000. Use a table to show the probability model. Policy holder outcome Payout x Probability P(x)

9. Copyright © 2010 Pearson Education, Inc. Slide 16 - 9 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.

10. Copyright © 2010 Pearson Education, Inc. Slide 16 - 10 Expected Value: Center (cont.) What is the expected value of an insurance policy payout?

11. Copyright © 2010 Pearson Education, Inc. Slide 16 - 11 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:

12. Copyright © 2010 Pearson Education, Inc. Slide 16 - 12 First Center, Now Spread… What is the variance for the insurance payouts? What is the standard deviation for the payouts? OutcomePayout xP(x)Deviation (x – μ)

13. Copyright © 2010 Pearson Education, Inc. A restaurant has a Valentine’s Day Special. The waiter brings the customers the four aces out of a deck and they choose a card. If they get the ace of hearts they deduct $20 from their meal cost. If they get a black ace, they pay the full amount. If they get the ace of diamonds they get to draw again. If this time they get the ace of hearts they win $10 off their meal. What is the probability of each outcome? What is the expected discount per couple? What is the variance and standard deviation for each couple? Slide 16 - 13

14. Copyright © 2010 Pearson Education, Inc. Your computer company has shipped 2 computers to your biggest customer. However, you now discover that they might have received refurbished instead of new computers. The computers were selected from a stock of 15, of which 4 were refurbished. If both computers are good, you’re good! If one is refurbished, then you will pay $100 to return it and send a new one. If both are refurbished, both will be sent back and you will lose the entire $1000 order. What is the expected value and standard deviation of the company’s loss? Slide 16 - 14

15. Copyright © 2010 Pearson Education, Inc. Slide 16 - 15 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) Example: Consider everyone in a company receiving a $5000 increase in salary.

16. Copyright © 2010 Pearson Education, Inc. Slide 16 - 16 More About Means and Variances (cont.) In general, multiplying each value of a random variable by a 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) Example: Consider everyone in a company receiving a 10% increase in salary.

17. Copyright © 2010 Pearson Education, Inc. Slide 16 - 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) 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)

18. Copyright © 2010 Pearson Education, Inc. A restaurant has a Valentine’s Day Special. The waiter brings the customers the four aces out of a deck and they choose a card. If they get the ace of hearts they deduct $20 from their meal cost. If they get a black ace, they pay the full amount. If they get the ace of diamonds they get to draw again. If this time they get the ace of hearts they win $10 off their meal. What is the probability of each outcome? What is the expected discount per couple? What is the variance and standard deviation for each couple? Slide 16 - 18

19. Copyright © 2010 Pearson Education, Inc. Slide 16 - 19 More About Means and Variances (cont.) When two couples dine on the same check at our restaurant, the restaurant doubles the amount of the discount. What are the new mean and standard deviation for two couples dining together?

20. Copyright © 2010 Pearson Education, Inc. Slide 16 - 20 More About Means and Variances (cont.) What if our two couples decide to get separate checks and share any winnings. Does it make any difference if they pay together or separately?

21. Copyright © 2010 Pearson Education, Inc. Slide 16 - 21 More About Means and Variances (cont.) Another restaurant has a competing program where customers can earn discounts on their meals. The manager says the average discount is $10 with a standard deviation of $15. How much more can you expect to win at this second restaurant, with what standard deviation?

22. Copyright © 2010 Pearson Education, Inc. Slide 16 - 22 More About Means and Variances (cont.) Our restaurant offers discounts with an average of $5.83 and a standard deviation of $8.62. The owner is planning to serve 40 couples on Valentine’s day. What is the expected total of discounts that will be given? What is the expected standard deviation of the total discounts?

23. Copyright © 2010 Pearson Education, Inc. Slide 16 - 23 Continuous Random Variables Random variables that can take on any value in a range of values are called continuous random variables. Now, any single value won’t have a probability, but… Continuous random variables have means (expected values) and variances. We won’t worry about how to calculate these means and variances in this course, but we can still work with models for continuous random variables when we’re given the parameters.

24. Copyright © 2010 Pearson Education, Inc. Slide 16 - 24 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.

25. Copyright © 2010 Pearson Education, Inc. Slide 16 - 25 Continuous Random Variables (cont.) The annual cost of medical care for dogs averages $100, with a standard deviation of $30, and for cats averages $120 with a standard deviation of $35. What is the expected difference in the cost of medical care for dogs and cats? What is the standard deviation of that difference? If the costs can be described by a normal model, what is the probability that medical expenses are higher for someone’s dog than for their cat?

26. Copyright © 2010 Pearson Education, Inc. Slide 16 - 26 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.

27. Copyright © 2010 Pearson Education, Inc. Slide 16 - 27 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.

28. Copyright © 2010 Pearson Education, Inc. Slide 16 - 28 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.

29. Copyright © 2010 Pearson Education, Inc. Slide 16 - 29 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.

30. Copyright © 2010 Pearson Education, Inc. Slide 16 - 30 What have we learned? (cont.) Normal models are once again special. Sums or differences of Normally distributed random variables also follow Normal models.