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Unformatted text preview: STAT 410 Fall 2008 Homework #4 (due Friday, September 26, by 3:00 p.m.) 1. Let X and Y have the joint probability density function f X, Y ( x , y ) = & & < < < + otherwise 1 4 x y y x a) Find f Y ( y ). f Y ( y ) = ( ) + 1 4 y dx y x = y y x x 1 2 4 2 + = 2 2 9 4 2 1 y y+ , 0 < y < 1. b) Find f Y  X ( y  x ). f X ( x ) = ( ) + x dy y x 4 = ( ) 2 2 x y y x + = 3 x 2 , 0 < x < 1. f Y  X ( y  x ) = 2 3 4 x y x + , 0 < y < x , 0 < x < 1. f Y  X ( y  x ) is undefined for x < 0 or x > 1. c) Find E ( Y  X ). E ( Y  X = x ) = + x dy x y x y 2 3 4 = 3 2 2 3 4 2 3 1 x y y x x + = 18 11 x , 0 < x < 1. E ( Y  X = x ) is undefined for x < 0 or x > 1. E ( Y  X ) = 18 X 11 . d) Are X and Y independent? Justify your answer. The support of ( X, Y ) is not a rectangle. & X and Y are NOT independent . OR f ( x , y ) f X ( x ) f Y ( y ). & X and Y are NOT independent . e) Find Cov ( X, Y ). E ( X ) = 1 2 3 dx x x = 4 3 . E ( X 2 ) = 1 2 2 3 dx x x = 5 3 . E ( Y ) = E [ E ( Y  X ) ] = E ( 18 X 11 ) = 4 3 18 11 = 24 11 . E ( X Y ) = E [ E ( X Y  X ) ] = E [ X E ( Y  X ) ] = E ( 18 X 11 2 ) = 5 3 18 11 = 30 11 . Cov ( X, Y ) = E ( X Y ) E ( X ) E ( Y ) = 24 11 4 3 30 11 = 480 11 . From the textbook: 2.3.3 Let f ( x 1 , x 2 ) = 3 2 2 1 21 x x , 0 < x 1 < x 2 < 1, zero elsewhere, be the joint pdf of X 1 and X 2 . (a) Find the conditional mean and variance of X 1 , given X 2 = x 2 , 0 < x 2 < 1. (b) Find the distribution of Y = E ( X 1  X 2 ). (c) Determine E ( Y ) and Var ( Y ) and compare these to E ( X 1 ) and Var ( X 1 ), respectively. f ( x 1 , x 2 ) = 3 2 2 1 21 x x , 0 < x 1 < x 2 < 1. (a) f 2 ( x 2 ) = & 2 1 3 2 2 1 21 x dx x x = 6 2 7 x , 0 < x 2 < 1. f 1  2 ( x 1  x 2 ) = 3 2 2 1 3 x x , 0 < x 1 < x 2 , 0 < x 2 < 1. E ( X 1  X 2 = x 2 ) = & 2 1 3 2 2 1 1 3 x dx x x x = 2 4 3 x , 0 < x 2 < 1. E ( X 1 2  X 2 = x 2 ) = & 2 1 3 2 2 1 2 1 3 x dx x x x = 2 2 5 3 x , 0 < x 2 < 1. Var ( X 1  X 2 = x 2 ) = 2 2 2 2 16 9 5 3 x x = 2 2 80 3 x , 0 < x 2 < 1....
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This note was uploaded on 10/22/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Fall '08 term at University of Illinois at Urbana–Champaign.
 Fall '08
 AlexeiStepanov
 Probability

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