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Unformatted text preview: Julie Novak, Tung Phan, and Wei Han Professor Rakhlin STAT 101 HW 6 Solutions October 31st, 2011 1. Chapter 10 Question 30 Question. #30: If percentage changes in the value of a stock are iid with mean 0, then how should we predict the change tomorrow if the change today was a 3 percent increase? How would the prediction change if the value today decreased by 2 percent? In either case, we should predict 0 for the change tomorrow, the expected value. Know- ing the change today has no effect on the distribution tomorrow if the changes are independent. 2. Chapter 10 Question 42 Question. #42: Two classmates both enjoy playing online poker. They both claim to win $ 300 on average when they play for an evening, even though they play at different sites on the Web. They do not always win the same amounts, and the SD of the amounts won is $ 100. (a) Identify the two random variables and summarize your assumptions Let X 1 and X 2 denote the winnings of the two players. We can assume these are independent since they play at different sites. (b) What is the mean and standard deviation of the total winnings made in a day by the two classmates? E ( X 1 + X 2 ) = 300 + 300 = $600 SD ( X 1 + X 2 ) = 100 2 + 100 2 = p 20 , 000 2 = $141 . 42 (c) Find the mean and standard deviation of the difference in the classmates winnings E ( X 1- X 2 ) = 300- 300 = $0 SD ( X 1- X 2 ) = 100 2 + 100 2 = p 20 , 000 2 = $141 . 42 (d) The classmates have decided to play in a tournament and are seated at the same virtual game table. How will this affect your assumptions about the random vari- ables? First, we may expect the quality of the other players to be better than usual, though this may not be the case. If so, then the mean winnings for each may fall and the variance increases. Second, by playing at the same table, for one to win means that the other must lose. This would lead to a negative correlation between X 1 and X 2 . 3. Chapter 10 Question 45 1 Question. #45: During the 2004-2005 NBA season, LeBron James of the Cleveland Cavaliers attempted 308 three-point baskets and made 108. He also attempted 1,376 two-point baskets and made 687 of these. Use these counts to determine the probabilities for the following questions. (a) Let a random variable X denote the result of a two-points attempt. X is either 0 or 2, depending on whether the basket is made. Find the expected value and variance of X. E ( X ) = P ( X = 0) 0 + P ( X = 2) 2 = 687 1376 2 = 0 . 9985 V ar ( X ) = E ( X- EX ) 2 = P ( X = 0) (0- EX ) 2 + P ( X = 2) (2- EX ) 2 = 1 (b) Let a second random variable Y denote the result of a three-point attempt. Find the expected value and variance of Y E ( Y ) = P ( Y = 0) 0 + P ( Y = 3) 3 = 108 308 3 = 1 . 0519 V ar ( Y ) = E ( Y- EY ) 2 = P ( Y = 0) (0- EY ) 2 + P ( Y = 3) (3- EY ) 2 = 2 . 0492 (c) In a game, LeBron attempts 5 three-point baskets and 20 two-point baskets. How many points do you expect him to score? (Hint: Use a collection of iid randommany points do you expect him to score?...
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