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test567-09sol

# test567-09sol - A-AE 567 Quiz Spring 2009 Clearly write...

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Unformatted text preview: A-AE 567 Quiz Spring 2009 Clearly write your final answer on the exam. NAME: 1 Problem 1. Assume that x and y are independent uniform random variables over the interval [0 , 1]. Let z be the random variable defined by z = | y- x | . Hint you may need F z ( z ). Then find (a) E z = 1 / 3. Because x and y are independent, f x , y ( x, y ) = f x ( x ) f y ( y ). In other words, f x , y ( x, y ) = 1 if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 = 0 otherwise . Using this we obtain E z = Z ∞-∞ Z ∞-∞ | y- x | f x , y ( x, y ) dydx = Z Z y ≥ x ( y- x ) f x , y ( x, y ) dxdy + Z Z x ≥ y ( x- y ) f x , y ( x, y ) dydx = Z 1 Z y ( y- x ) dxdy + Z 1 Z x ( x- y ) dydx = 2 Z 1 Z y ( y- x ) dxdy = 2 Z 1 yx- x 2 2 ¶fl fl fl fl y x =0 dy = Z 1 y 2 dy = 1 3 . (b) This is the meeting problem done in the notes with L = 1. Hence f z ( z ) = 2(1- z ) if 0 ≤ z ≤ 1 = 0 otherwise . 2 One can also use f z ( z ) to compute E z . To see this observe that E z = Z ∞-∞ zf z ( z ) dz = 2 Z 1 z (1- z ) dz = 2 1 2- 1 3 ¶ = 1 3 ....
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test567-09sol - A-AE 567 Quiz Spring 2009 Clearly write...

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