hw08_sol

hw08_sol - Homework 8 November 2 2007 Problem 1 a f U,V u,v...

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Unformatted text preview: Homework 8 November 2, 2007 Problem 1. a) f U,V ( u,v ) = 1 . 5 = 2 . ( ≤ U ≤ 1 , ≤ V ≤ U ) f V | U ( v | u ) = f V,U ( v,u ) f U ( u ) = 2 2 u = 1 u , ( ≤ U ≤ 1 , ≤ V ≤ U ) E [ V | U ] = R u v 1 u dv = v 2 2 u | u = u 2 f V ( v ) = R 1 v 2 du = 2- 2 v E [ V ] = R 1 v (2- 2 v ) dv = v 2- 2 / 3 v 2 | 1 = 1 / 3 b) f X ( x ) = 1 , ( ≤ X ≤ 1 ) f Y | X ( y | x ) = 1 x , ( ≤ X ≤ 1 , ≤ Y ≤ X ) E [ Y | X ] = R x y 1 x dy = y 2 2 x = x 2 f X,Y ( x,y ) = f Y | X ( y | x ) f X ( x ) = 1 x f Y ( y ) = R 1 y 1 x dy =- lny E [ Y ] = R 1 y (- lny ) dy =- y 2 4 (2 lny- 1) | 1 = 1 4 c) It is because f X ( x ) are di erent in part a) and part b). Therefore, even though the region and the conditional distribution are the same, the marginal distributions of Y are di erent. Problem 2. Let X= the lifetime of the lightbulb. Y=The lament of the lightbulb. Y = 1 if it has a good lament, Y = 0 if it has a bad lament P ( Y = 1) = 0 . 95 , E ( X | Y = 1) = 600 , E ( X | Y = 0) = 50 E [...
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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.

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hw08_sol - Homework 8 November 2 2007 Problem 1 a f U,V u,v...

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