Fat Chance  Homework 9


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1 Fat Chance  Homework 9 December, 201 Due before class on Wednesday, November 10. This is the first part of the assignment. Contact Eric Riedl, with questions. These problems were posted after Wednesday s lecture. 1. Suppose you roll two dice for each of three games. In game X, you get $1 for each 6, in game Y, you get $2 for each 6, and in game Z, you get $11 if you get 2 6 s (but no money for only one 6). (a) Compute the expected value and variance for game X. (Solution) The expected value is = = 1 3. The variance is (0 1 3 ) (1 1 3 ) (2 1 3 )2 = = = (b) Notice that game Y is the same as game X, except with the payoffs doubled. Compute the expected value and variance for game Y. How do they change when you double the payoffs? (Solution) The expected value is = 2 3. The variance is (0 2 3 ) (2 2 3 ) ( 2 3 )2 = 0 36 = The expected value doubles while the variance quadruples. (c) Compute the expected value and variance for game Z. How do they compare to game X? When you compare the variances, does this match your intuition? (Solution) The expected value for Z is = 36, about the same as for game X. The variance for Z is (0 36 ) (11 36 )2 3.27, much bigger than the variance for game X. This makes sense, because there s a lot more variation in outcomes in game Z. 2. Suppose you roll five dice. If s is the number of 5 s you roll, then you win 2 s dollars. (a) What are the outcomes, probabilities and payoffs in this game? (Solution) We make the following chart. outcome zero 5 s one 5 two 5 s three 5 s four 5 s five 5 s 5 probability payoff ( 5 1)5 ( 5 2)5 3 ( 5 3)5 2 ( 5 )
2 (b) What is the expected value of this game? (Solution) We just multiply the probabilities and payoffs and then sum to get (c) What is the variance of (the payouts in) this game? (Solution) We subtract the expected payout from the actual payout, multiply by the probability, then sum, to get about Consider the games A and B. The probabilities and outcomes of both games are the same. In each game, you flip four coins, and the outcomes are the different numbers of heads. In game A, you get $1 per head. In game B, you get $2 h, where h is the number of heads. (a) What is the expected value of game A? (Solution) The expected value is = 2. (b) What is the expected value of game B? Is it the same as 2 expected value of game A? (Careful on this one; you ll want to compute both quantities out using the definitions.) (Solution) The expected value is = which is different from 2 expected value of game A. (c) What is the variance of game A? (Solution) 1 (0 2)2 + (1 2)2 + 6 (2 2)2 + (3 2)2 + 1 ( 2)2 = 1 (d) What is the variance of game B? Does it match with your intuition that the variance is higher or lower than that of game A? (Solution) The variance is 1 81 (1 ) (2 ) ( ) (8 ) ( ) It makes sense that it is higher than that of game A, because there is more variation in payoffs. These questions were posted after Friday s lecture.. Suppose you are playing ChuckaLuck (which we will call game C), where you roll three dice and get payoffs depending on how many 6 s you roll. Zero 6 s gives you zero dollars. One or two 6 s gives you $2. Three 6 s gives you $10. (a) What is the expected value of C? (Solution) The expected value is = (b) Suppose you want to increase or decrease all of the payouts by the same number, so that the expected value of the game is 0. What would the payouts of this new game, C, be? (Solution) Subtract.88 from all of the payouts to get.88 with zero 6 s, 1.12 with one or two 6 s, and 9.12 with three sixes. 2
3 (c) What is the variance of C? What is the variance of C? What are the standard deviations? (Solution) The variance of C is (0.88) (2.88) (10.88) It is the same as the variance of C (since subtracting a constant doesn t affect the variance). (d) Suppose you want to multiply all the payouts by the same number to get a game C with expected value 0 and variance 1. What are the payouts of C? (Solution) We divide by the standard deviation 1.36 to get payouts of.756,.96, and Suppose you have an unfair coin that comes up heads with probability 1/, and you toss it three times. You get $2 for every heads. (a) What is the expected value of this game? Is it the same as three times the expected value of tossing a single coin? (Solution) This will be three times the expected value of tossing a single coin, i.e., 1.5. (b) What is the variance of this game? Is it the same as three times the variance of tossing a single coin? (Solution) The variance will also be three times the variance of tossing a single coin, or (c) What is the standard deviation of this game? (Solution) The standard deviation is the square root of the variance, i.e., 1.5. (d) What is the normalized form of this game? (i.e., how do you modify this game by adding constants and multiplying/dividing by constants so that the expected value is 0 and the standard deviation is 1)? The normalized form is given by subtracting 1.5 and then dividing by 1.5, for payouts of 1, 1 3, 5 3, and Suppose you roll a sided die, and get paid the number of dollars on the die. (a) What is the normalized form of this game? (Solutions) The expected value is = 2.5. The variance is 1 ((1 2.5)2 + (2 2.5) 2 + (3 2.5) 2 + ( 2.) 2 ) = The standard deviation is 1.25 = Thus, you get the normalized form by subtracting 2.5 and dividing by 1.118, for payouts of 1.32,.7,.7, and (b) What is the expected value and variance of the normalized form of this game? (Here, please work it out explicitly, even if you think you already know what they will be). Is this what you were expecting? (Solutions) The expected value is = 0, and the variance is ( 1.32)2 +(.7) 2 +(.7) 2 +(1.32) 2 = 1, as expected. 3
4 7. Suppose you roll sided dice and get paid the number on the dice. (a) What is the expected value of this game? (Solution) The expected value of rolling one die is 3.5, so the expected value of rolling 100 dice is 350. (b) What is the variance of this game? (Solution) The variance of rolling one die is , so the variance or rolling 100 dice is (c) Suppose you want to relabel the numbers on the die by subtracting the same number from each face so that the expected value is 0. What numbers should you put on the die? We can subtract the expected value 3.5, to get 2.5, 1.5,.5,.5, 1.5, and 2.5 as the numbers on our die. (d) Suppose further that you want to divide all the numbers on the die by the same constant so that the variance of rolling 100 dice is 1. Now what numbers should you put on the dice? (Solution) Note that this question is not asking us to normalize the game of rolling one die. We want the variance of rolling 100 dice to be 1, so we want the variance of rolling one die to be.01. Thus, we should divide by ten times the standard deviation of rolling one die (i.e. divide by 17.08), so that the numbers on the dice are.16,.0878,.0293,.0293,.0878 and.16. These problems were posted after Monday s lecture. longer; they count double. They re a little 8. Consider the game G where you flip a (fair) coin and get paid $1 if it s heads and $0 if it s tails. Let H = G(10) be the game G + G + G + G + G + G + G + G + G + G; that is, you flip 10 coins and are paid $1 for each head. (a) What are the expected value and variance of H? (Solution) The expected value of G is 1 2, so the expected value of H will just be 10 2 = 5. Similarly, the variance of G is 1, so the variance of H will be 2.5. (b) What is the normalized form H 0 of the game H? (Solution) This will just be H (c) Draw a bar chart of the game H, labeling the heights of the various
5 (Solution) (d) Draw a bar chart of the game H 0, labeling the heights of the various (Solution) 9. Now we ll do another problem along the lines of the last one, but starting with a different game. Let J be the game where you roll a fair die, and collect $1 if it s a 6 and $0 otherwise. Let K = J(10) be the game where you roll 10 dice and are paid $1 for each 6. (a) What are the expected value and variance of K? (Solution) The expected value of J is 1 6, so the expected value of K will be 10 6 = frac53. The variance of J is = 5 36, so the variance of K will be = (b) What is the normalized form K 0 of the game K? (Solution) This is simply K
6 (c) Draw a bar chart of the game K, labeling the heights of the various (Solution) (d) Draw a bar chart of the game K 0, labeling the heights of the various (Solution) (e) How does the bar chart of K 0 compare to the bar chart for H 0 you drew in part (d) of the last problem? (Solution) It looks broadly similar, but not symmetrical like the graph in the last problem. It is not as good an approximation of the normal distribution. 6
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