prelim2soln

prelim2soln - ORIE 3500/5500 Fall ’10 Prelim 2 Solution...

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Unformatted text preview: ORIE 3500/5500, Fall ’10 Prelim 2 Solution Prelim 2 Solution Problem 1 (a) EX = Z 1 Z y- 1 x 2 3 dxdy =- 2 9 , EX 2 = Z 1 Z y- 1 x 2 2 3 dxdy = 5 18 , Var X = EX 2- ( EX ) 2 = 37 162 , EY = Z 1 Z y- 1 y 2 3 dxdy = 5 9 , EY 2 = Z 1 Z y- 1 y 2 2 3 dxdy = 7 18 , Var Y = EY 2- ( EY ) 2 = 13 162 . (b) E ( XY ) = Z 1 Z y- 1 xy 2 3 dxdy =- 1 12 , Cov( X,Y ) = E ( XY )- ( EX )( EY ) = 13 324 , ρ X,Y = Cov( X,Y ) √ Var X √ Var Y = . 296 . (c) Var( aX + Y ) = a 2 Var X + 2 a Cov( X,Y ) + Var Y. This is a quadratic function of a with a positive leading coefficient, so its mini- mum is achieved at the point where the derivative vanishes. Taking a derivative and setting it equal to zero: 2 a Var X + 2Cov( X,Y ) = 0 gives a =- Cov( X,Y ) Var X =- 13 74 . Problem 2 (a) P ( X ≤ Y ) = P ( X = min( X,Y )) = λ X λ X + λ Y = 1 5 1 5 + 1 8 = 8 13 P ( X > Y ) = 1- 8 13 = 5 13 1 ORIE 3500/5500, Fall ’10 Prelim 2 Solution ⇒ E( I ) = 1 · 8 13 + 2 · 5 13 = 18 13 (b) Let Z = min( X,Y ) then Z ∼...
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prelim2soln - ORIE 3500/5500 Fall ’10 Prelim 2 Solution...

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