410Hw04ans - STAT 410 Fall 2009 Homework #4 (due Friday,...

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STAT 410 Fall 2009 Homework #4 (due Friday, September 25, by 3:00 p.m.) 1. Let X, Y, and Z be i.i.d. Uniform [ 0 , 1 ] random variables Find the probability distribution of W = X + Y + Z. That is, find ( 29 w f W . Hint: If V = X + Y, we know the p.d.f. of V, f V ( v ) ( see Examples for 09/18/2009 ): f V ( v ) = v if 0 < v < 1, f V ( v ) = 2 – v if 1 < v < 2, f V ( v ) = 0 otherwise. Now use convolution formula to find the p.d.f. of W = V + Z. There will be 5 possible cases; two of them are “boring”, two of them are “exciting”, and one is “really exciting”. ( 29 < < = otherwise 0 1 0 1 Z z z f ( 29 < < - = < - < = - otherwise 0 1 1 otherwise 0 1 0 1 Z w v w v w v w f ( 29 w f W = ( 29 ( 29 ( 29 - = - + dv v w f v f w f Z V Z V (convolution) Case 1 : w < 0. ( 29 w f W = 0.
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Case 2 : 0 < w < 1. Then w – 1 < 0. ( 29 ( 29 2 1 2 0 W w dv v w f w = = . Case 3 : 1 < w < 2. Then 0 < w – 1 < 1. ( 29 ( 29 ( 29 ( 29 2 3 3 1 2 1 2 1 1 1 W - + - = - + = - w w dv v dv v w f w w ( 29 ( 29 2 1 2 1 + - - = w w . Case 4 : 2 < w < 3. Then 1 < w – 1 < 2. ( 29 ( 29 ( 29 ( 29 2 3 2 9 3 2 1 2 2 2 2 1 W w w w dv v w f w - = + - = - = - . Case 5 : w > 3. Then w – 1 > 2. ( 29 w f W = 0.
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2. Let X and Y be two independent random variables, with probability density functions f X ( x ) and f Y ( y ) , respectively. ( 29 = otherwise 0 1 0 3 2 X x x x f ( 29 = otherwise 0 1 0 2 Y y y y f Find the p.d.f. f W ( w ) of W = X + Y. f W ( w ) = ( 29 ( 29 - - dx x w f x f Y X . ( 29 x w f - Y = ( 29 - - otherwise 0 1 0 if 2 x w x w = ( 29 - - otherwise 0 1 if 2 w x w x w Case 1 . w < 0. ( 29 w f Y X + = 0. Case 2 . 0 < w < 1. ( 29 w f Y X + = ( 29 - w dx x w x 0 2 2 3 = 0 4 3 2 3 2 = = - x w x x w x = 4 2 1 w .
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Case 3 . 1 < w < 2. ( 29 w f Y X + = ( 29 - - 1 1 2 2 3 w dx x w x = 1 1 4 3 2 3 2 - = = - w x x x w x = ( 29 ( 29 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. ( 29 w f Y X + = 0. ( 29 w f Y X + = < < - + - < < otherwise 0 2 1 2 3 2 1 1 0 2 1 2 4 4 w w w w w w
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OR Case 1 . w < 0. F X + Y ( w ) = 0. ( 29 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 = … ( 29 w f Y X + = F X ' + Y ( w ) = … Case 3 . 1 < w < 2. F X + Y ( w ) = … ( 29 w f Y X + = F X ' + Y ( w ) = … Case 4 . w > 2. F X + Y ( w ) = 1. ( 29 w f Y X + = F X ' + Y ( w ) = 0.
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3. Let X and Y have the pdf f ( x , y ) = 1, 0 < x < 1, 0 < y < 1, zero elsewhere. Find the
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410Hw04ans - STAT 410 Fall 2009 Homework #4 (due Friday,...

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