HW07 - STAT 36-217 HW 7 due Thursday 10:30 AM PLEAE USE...

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STAT 36-217, HW 7, due Thursday 10/22/2009, 10:30 AM PLEAE USE THIS AS THE COVER PAGE Your Name: 1
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1. Let X and Y be independent random variables with means μ x and μ y and variances σ 2 x and σ 2 y . Show that Var( XY ) = σ 2 x σ 2 y + μ 2 y σ 2 x + μ 2 x σ 2 y . 2. Let X and Y be independent normal random variables, each having mean μ and variance σ 2 . Calculate Cov( X + Y, X - Y ). 3. The joint density of X and Y is f ( x, y ) = y 2 - x 2 8 e - y , 0 < y < , - y x y. Show that E [ X | y = y ] = 0. 4. We start with a stick of length = 1 . We break it at a point which is chosen according to a uniform distribution and keep the piece, of length Y , that contains the left end of the stick. We then repeat the same process on the piece that we were left with, and let X be the length of the remaining piece after breaking for the second time. Find the joint PDF of Y and X , and the marginal PDF of X . 5. A total of 11 people, including you, are invited to a party. The time at which peo- ple arrive at the party are independent uniform random variables U(0,1). Find the
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