quiz2sol - PROBABILITY AND DECISION MAKING 45-730 QUIZ...

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PROBABILITY AND DECISION MAKING 45-730 QUIZ 2 (OFFLINE) Problem 1 ( Expectation and Variance ) . You and your friend play the following game. You toss a six sided dice. Your score, denoted by X, is the number that appears on the dice and your friend’s score, denoted by Y is 6 minus the number that appears on the dice. The person who gets the higher score wins the game. In case both scores are equal, the game results in a tie. (1.1) What is your expected score E[X]? Solution. E [ X ] = 6 i =1 1 6 · i = 3 . 5 (1.2) What is the expected score of your friend E[Y]? Solution. Y = 6 - X . Therefore, E [ Y ] = E [6 - X ] = 6 - E [ X ] = 2 . 5 (by linearity of expectation). (1.3) What is Var[X]? Solution. V ar [ X ] = 6 i =1 1 6 · ( i - E [ X ]) 2 = 2 . 92. (1.4) What is Var[Y]? Solution. V ar [ Y ] = V ar [6 - X ] = V ar [ X ] = 2 . 92. (1.5) Let Z be a random variable such that Z = X - Y . What is E[Z]? Solution. Using linearity of expectation, E [ Z ] = E [ X ] - E [ Y ] = 1. (1.6) What is the value of Var[Z]? Solution. Note that Z = x - (6 - X ) = 2 X - 6. Therefore, V ar [ Z ] = 4 · V ar [ X ] = 11 . 67. (1.7) What is the probability that Z is greater than zero? Note that this is the probability that you win the game. Solution. Z > 0 (2 X - 6) > 0 X > 3. Thus, P ( Z > 0) = 1 2 . (1.8) Now, suppose you play the following modified game. You and your friend both toss a six sided dice independently and let X denote the number that you get and Y denote the number that your friend gets. Again, let Z = X - Y . What is the value of E[Z] and Var[Z]? Solution. From part 1, we know E [ X ] = 3 . 5. Since, X are Y are i.i.d, E [ X ] = E [ Y ] = 3 . 5. Therefore, E [ Z ] = 0 and V ar [ Z ] = V ar [ X ] + V ar [ Y ] = 5 . 83. Problem 2. You and your friend play the following game. You toss a six sided dice and the number that appears on the dice is your score, denoted by X. Your friend tosses a fair coin six times and his score is the number of times the toss resulted in a heads, denoted by Y. The person getting a higher score wins. In case both scores are equal, the game results in a tie. (2.1) What is the probability that the game will result in a tie, i.e. P ( X = Y )? 1
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2 QUIZ 2 (OFFLINE) Solution. For i = 1 , . . . , 6, P ( X = i, Y = i ) = 1 6 · C 6 i 2 6 . Note that C 6 0 = 1, C 6 1 = 6, C 6 2 = 15, C 6 3 = 20, C 6 4 = 15, C 6 5 = 6, C 6 6 = 1. Thus, P (game results in a tie) = 6 i =1 P ( X = i, Y = i ) = 1 6 6 i =1 C 6 i 2 6 = 1 6 · 6 + 15 + 20 + 15 + 6 + 1 64 = 1 6 · 63 64 . (2.2) What is the probability that you win the game, i.e. P ( X > Y )? Solution. P ( X > Y ) = 5 i =0 P ( Y = i, X > i ) = 5 i =0 C 6 i 2 6 · 6 - i 6 = 1 2 6 · 5 i =0 C 6 i - 1 6 · 2 6 · 5 i =0 i · C 6 i = (1 + 6 + 15 + 20 + 15 + 6) 64 - (0 · 1 + 1 · 6 + 2 · 15 + 3 · 20 + 4 · 15 + 5 · 6) 6 · 64 = 63 64 - 31 64 = 1 2 .
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