hw05_sol - Homework 5 October 10, 2007 Problem 1. Denote...

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Homework 5 October 10, 2007 Problem 1. Denote the variance as g ( a ) . g ( a ) Var ( aX + (20000 - a ) Y ) = a 2 σ 2 X + (20000 - a ) 2 σ 2 Y + 2 a (20000 - a ) ρσ X σ Y dg ( a ) da = 2 2 X - 2(20000 - a ) σ 2 Y + 2(20000 - a ) ρσ X σ Y - 2 aρσ X σ Y = 0 Solving this equation, we have a = 20000( σ 2 Y - ρσ X σ Y ) σ 2 X + σ 2 Y - 2 ρσ X σ Y If a is chosen in such a way, the resulting variance is minimized because d 2 g ( a ) da 2 = 2( σ 2 X + σ 2 Y - 2 ρσ X σ Y ) 2( σ 2 X + σ 2 Y - 2 σ X σ Y ) = 2( σ X - σ Y ) 2 0 . Problem 2. a) E ( XY ) = R 1 0 R 1 - x - (1 - x ) 3 x 2 ydydx = R 1 0 3 x 2 2 y 2 | 1 - x - (1 - x ) dx = 0 f X ( x ) = R 1 - x - (1 - x ) 3 xdy = 6 x (1 - x ) = 6 x - 6 x 2 μ = EX = R 1 0 6 x 2 - 6 x 3 dx = 2 x 3 | 1 0 - 3 2 x 4 | 1 0 = 1 2 f Y ( y ) = R 1 -| y | 0 3 xdx = 3(1+ y 2 - 2 | y | ) 2 ν = EY = R 0 - 1 3 2 ( y + y 3 + 2 y 2 ) dy + R 1 0 3 2 ( y + y 3 - 2 y 2 ) dy = 0 Cov ( X,Y ) = E ( X · Y ) - μν = 0 - 0 = 0 b) X and Y are not independent. Since f X ( x ) · f Y ( y ) 6 = f X,Y ( x,y ) 1
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This note was uploaded on 11/12/2008 for the course ORIE 360 taught by Professor Ehrlichman during the Fall '07 term at Cornell University (Engineering School).

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hw05_sol - Homework 5 October 10, 2007 Problem 1. Denote...

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