09_12_1 - STAT 410 Examples for 09/12/2011 (part 1) Fall...

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Examples for 09/12/2011 (part 1) Fall 2011 2. Let the joint probability density function for ( X , Y ) be ( ) + = otherwise 0 1 , 1 0 , 1 0 60 , 2 y x y x y x y x f Recall: f X ( x ) = ( ) 2 2 1 30 x x - , 0 < x < 1, E ( X ) = 2 1 , Var ( X ) = 252 9 , E ( X 2 ) = 252 72 = 7 2 ; f Y ( y ) = ( ) 3 1 20 y y - , 0 < y < 1, E ( Y ) = 3 1 , Var ( Y ) = 252 8 , E ( Y 2 ) = 252 36 = 7 1 ; f Y | X ( y | x ) = ( ) ( ) x f y x f , X = ( ) 2 1 2 x y - , 0 < y < 1 – x , 0 < x < 1. f X | Y ( x | y ) = ( ) ( ) y f y x f , Y = ( ) 3 2 1 3 y x - , 0 < x < 1 – y , 0 < y < 1. E ( Y | X = x ) = ( ) - - x dy x y y 1 0 2 1 2 = ( ) 1 3 2 x - , 0 < x < 1. E ( Y | X ) = ( ) X 1 3 2 - . E ( E ( Y | X ) ) = 3 1 = E ( Y ). E [ ( E ( Y | X ) ) 2 ] = ( ) ( ) ( ) 2 X E X E 2 1 9 4 + - = + - 7 2 2 1 2 1 9 4 = 63 8 . Var ( E ( Y | X ) ) = 2 3 1 63 8 - = 63 1 . OR
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This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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09_12_1 - STAT 410 Examples for 09/12/2011 (part 1) Fall...

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