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Unformatted text preview: STAT 409 Homework #1 Fall 2009 (due Friday, September 4, by 4:00 p.m.) 1. Let Î» > 0 and let X be a random variable with the probability density function f ( x ) = 1 Î» Î» + x , x > 1, zero otherwise. Let W = ln ( X ). What is the probability distribution of W ? w = ln ( x ) x = g â€“ 1 ( w ) = e w dw dx = e w x > 1 â‡’ w > 0 f W ( w ) = f X ( g â€“ 1 ( w ) ) dw dx = 1 Î» Î» w w e e â‹… + = w e Î» Î» , w > 0. OR F X ( x ) = âˆ« + x dy y 1 1 Î» Î» = Î» 1 1 x ., x > 1. F Y ( y ) = P ( W â‰¤ w ) = P ( X â‰¤ e w ) = F X ( e w ) = w e Î» 1 , w > 0. f W ( w ) = w e Î» Î» , w > 0. W has Exponential distribution with mean 1 / Î» . 2. Let X be a Uniform ( 0, 1 ) and Y be a Uniform ( 0, 3 ) independent random variables. Let W = X + Y. Find and sketch the p.d.f. of W. ( 29 < < = otherwise 1 1 X x x f ( 29 < < = otherwise 3 3 1 Y y y f F W ( w ) = P ( W â‰¤ w ) = P ( X + Y â‰¤ w ) = â€¦ Case 1 : 0 < w < 1. Case 2 : 1 < w < 3. Case 3 : 3 < w < 4. â€¦ = âˆ« âˆ«  â‹… w x w dx dy 3 1 1 â€¦ = âˆ« âˆ«  â‹… 1 3 1 1 dx dy x w â€¦ = âˆ« âˆ«  â‹… 1 3 1 3 1 1 1 w x w dx dy = 2 6 1 w . = ( 29 1 2 6 1 w . = ( 29 2 4 6 1 1 w . f W ( w ) = F W ' ( w ) = â€¦ â€¦ = w 3 1 , â€¦ = 3 1 , â€¦ = ( 29 4 3 1 w , 0 < w < 1. 1 < w < 3. 3 < w < 4. OR ( 29 ( 29 ( 29 âˆ« âˆž âˆž + â‹… = dx x w f x f w f Y X Y X ( 29 < < = otherwise 3 3 1 Y y y f ( 29 < < < < = = otherwise 3 3 1 otherwise 3 3 1 Y w x w x w x w f Case 1 : 0 < w < 1. f W ( w ) = âˆ« â‹… w dx 3 1 1 = w 3 1 . Case 2 : 1 < w < 3. f W ( w ) = âˆ« â‹… 1 3 1 1 dx = 3 1 . Case 3 : 3 < w < 4. f W ( w ) = âˆ« â‹… 1 3 3 1 1 w dx = ( 29 4 3 1 w . OR ( 29 ( 29 ( 29 âˆ« âˆž âˆž + â‹… = dy y f y w f w f Y X Y X ( 29 < < = otherwise 1 1 X x x f ( 29 < < = < < = otherwise 1 1 otherwise 1 1 X w y w y w y w f Case 1 : 0 < w < 1. f W ( w ) = âˆ« â‹… w dy 3 1 1 = w 3 1 . Case 2 : 1 < w < 3. f W ( w ) = âˆ« â‹… w w dy 1 3 1 1 = 3 1 ....
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This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.
 Spring '11
 Stepanov
 Probability

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