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

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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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. 0 < w < 1. ( ) w f Y X + = ( ) & - w dx x w x 0 2 2 3 = 0 4 3 2 3 2 = = ± ² ³ ´ µ - 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 - = = ± ² ³ ´ µ - 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
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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.

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

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