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416-2sol

# 416-2sol - Denker SPRING 2010 416 Stochastic Modeling...

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Denker SPRING 2010 416 Stochastic Modeling - Assignment 2 SOLUTIONS Problem 1: E [ X | Y = y ] means the expectation of the distribution of X when the outcome y of the random variable Y is known. First we calculate f X | Y ( x | y ). f Y ( y ) = integraldisplay y - y y 2 - x 2 8 e - y dx = 1 6 y 3 e - y . f X | Y ( x | y ) = y 2 - x 2 8 e - y 1 6 y 3 e - y = 3 4 ( 1 y - x 2 y 3 ) . E [ X | Y = y ] = integraldisplay y - y x 3 4 ( 1 y - x 2 y 3 ) dx = 0 , since the integrand is point symmetric around 0. Problem 2: Let N A (resp. N B ) denote the number of plays needed to win two in a row when beginning to play with A (resp. B). Let X 1 , X 2 , ... denote the sequence of independent random variables of the game when beginning with opponent A. Define a random variable Y by Y = 1, if X 1 = 0, Y = 2, if X 1 = 1 , X 2 = 1, and Y = 3, if X 1 = 1 , X 2 = 0. Then P ( Y = 1) = 1 - p A , P ( Y = 2) = p A p B and P ( Y = 3) = p A (1 - p B ) and E [ N A ] = E [ N A | Y = 1] P ( Y = 1) + E [ N A | Y = 2] P ( Y = 2) + E [ N A | Y = 3] P ( Y = 3) . If Y = 1, you plaed once with no win, so you start the same game playing B first. So the distribution of N A given Y = 1 equals that of 1 + N B , so E [ N A | Y = 1] = 1 + E [ N B ] .

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416-2sol - Denker SPRING 2010 416 Stochastic Modeling...

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