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# HW07 Answers - Solution for 36217 Wanjie Wang Teacher...

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Solution for 36217 Wanjie Wang Teacher: Jiashun Jin October 22, 2009 If there is any question, please contact me at [email protected] 1. (10 Points) Let X and Y be independent random variables... E [ Y 2 ] = V ar ( Y ) + ( EY ) 2 = σ 2 y + μ 2 y Because X and Y are independent, V ar ( XY ) = V ar ( E [ XY | Y ]) + E [ V ar ( XY | Y )] = V ar ( Y E [ X | Y ]) + E [ Y 2 V ar ( X | Y )] = V ar ( Y μ x ) + E [ Y 2 σ 2 x ] = μ 2 x σ 2 y + σ 2 x ( σ 2 y + μ 2 y ) = σ 2 x σ 2 y + μ 2 y σ 2 x + μ 2 x σ 2 y 2. (10 Points) Calculate Cov( X + Y , X - Y )... Because X and Y are independent, Cov( X , Y )=0. Cov ( X + Y, X - Y ) = Cov ( X, X ) - Cov ( Y, Y ) + Cov ( X, Y ) - Cov ( Y, X ) = σ 2 - σ 2 = 0 3. (10 Points) The joint density of X and Y is f ( x, y ) = y 2 - x 2 8 e - y ... E [ X | Y = y ] = Z y - y x y 2 - x 2 8 e - y dx = y 2 e - y 8 Z y - y xdx - e - y 8 Z y - y x 3 dx = 0 - 0 = 0 4. (10 Points) Stick problem. 1

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Obviously, Y Uniform(0,1), X | Y Uniform(0, Y ). So, f ( x, y ) = 1 × 1 /y = 1 /y, 0 < x y, 0 y 1. f X ( x ) = R 1 0 1 /yI (0 ,y ) ( x ) dy = R 1 x 1 /ydy = log y | 1 x = - log x 5. (10 Points) Arriving time. Let Y denotes my arriving time. Then Y Uniform (0 , 1).
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HW07 Answers - Solution for 36217 Wanjie Wang Teacher...

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