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Exam1_2ans - STAT 410 Spring 2008 Name ANSWERS Exam 1 part...

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STAT 410 Spring 2008 Name ANSWERS . Exam 1 ( part 2 ) ( 4 points ) No credit will be given without supporting work. If the answer is a function, its support must be included. 1. (1) Let X and Y be two independent random variables, with probability density functions f X ( x ) and f Y ( y ) , respectively. ( ) ° ± ° ² ³ = otherwise 0 1 0 3 2 X x x x f ( ) ± ² ³ = otherwise 0 1 0 2 Y y y y f Find the p.d.f. f W ( w ) of W = X + Y. ( ) x w f - Y = ( ) ( ) ´ - - dx x w f x f Y X . ( ) x w f - Y = ( ) ± ² ³ - - otherwise 0 1 0 if 2 x w x w = ( ) ± ² ³ - - otherwise 0 1 if 2 w x w x w Case 1 . w < 0. ( ) w f Y X + = 0.
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Case 2 . 0 < w < 1. ( ) w f Y X + = ( ) ´ - w dx x w x 0 2 2 3 = 0 4 3 2 3 2 = = µ · ¸ ¹ º - x w x x w x = 4 2 1 w . Case 3 . 1 < w < 2. ( ) w f Y X + = ( ) ´ - - 1 1 2 2 3 w dx x w x = 1 1 4 3 2 3 2 - = = µ · ¸ ¹ º - w x x x w x = ( ) ( ) 4 3 1 2 3 1 2 2 3 2 - + - - - w w w w = w w w 2 3 2 1 2 4 - + - . Case 4 . w > 2. ( ) w f Y X + = 0.
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OR Case 1 . w < 0. F X + Y ( w ) = 0. ( ) w f Y X + = F X ' + Y ( w ) = 0. Case 2 . 0 < w < 1. F X + Y ( w ) = ´ ´ - w x w dx dy y x 0 0 2 2 3 = … ( ) w f Y X + = F X ' + Y ( w ) = … Case 3 . 1 < w < 2. F X + Y ( w ) = … ( ) w f Y X + = F X ' + Y ( w ) = … Case 4 . w > 2. F X + Y ( w ) = 1. ( ) w f Y X + = F X ' + Y ( w ) = 0.
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( ) w f Y X + = ° ° ° ± ° ° ° ² ³ < < - + - < < otherwise 0 2 1 2 3 2 1 1 0 2 1 2 4 4 w w w w w w 2. (2) Let X and Y have the joint probability density function f X , Y ( x , y ) = x 1 , x > 1, 0 < y < x 1 , zero elsewhere. a) Find f Y ( y ) . f Y ( y ) = ´ y dx x 1 1 1 = ( ) 1 1 ln y x = 1 1 ln ln - y = – ln y , 0 < y < 1.
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b) Find f Y | X ( y | x ) . f X ( x ) = ´ x dy x 1 0 1 =
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