Hw09ans - STAT 408 Spring 2009 Homework#9(due Friday April...

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STAT 408 Spring 2009 Homework #9 (due Friday, April 3, by 3:00 p.m.) 1. Suppose that the random variables X and Y have joint p.d.f. f ( x , y ) given by f ( x , y ) = C x 2 y , 0 < x < y , x + y < 2. a) Sketch the support of ( X , Y ). That is, sketch { 0 < x < y , x + y < 2 }. b) What must the value of C be so that f ( x , y ) is a valid joint p.d.f.? Must have ( 29 ∫ ∫ - - dy dx y x f , = 1. - 1 0 2 2 dx dy y x x x C = = - = 1 0 2 2 2 2 dx y x x y x y C = ( 29 [ ] - - 1 0 2 2 2 2 2 dx x x x C = ( 29 - 1 0 3 2 2 2 dx x x C C = 0 1 4 3 2 3 2 - x x C C = 6 C = 1. C = 6 .
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c) Find P ( X + Y < 1 ). - 5 . 0 0 1 2 6 dx dy y x x x = ( 29 = - = 5 . 0 0 1 2 2 3 dx y x x y x y = ( 29 [ ] ( 29 - - 5 . 0 0 2 2 2 1 3 dx x x x = ( 29 - 5 . 0 0 3 2 6 3 dx x x = 0 5 . 0 4 3 2 3 - x x = 4 3 2 1 2 3 2 1 - = 32 3 8 1 - = 32 1 = 0.03125 . d) Find the marginal probability density function for X. First, X can only take values in ( 0 , 1 ). f X ( x ) = ( 29 - , dy y x f = - x x dy y x 2 2 6 = ( 29 3 2 2 2 x y x y y x = - = = ( 29 { } 2 2 2 2 3 x x x - - = 12 x 2 – 12 x 3 = 12 x 2 ( 1 – x ), 0 < x < 1.
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e) Find the marginal probability density function for Y. First, Y can only take values in ( 0 , 2 ). f Y ( y ) = ( 29 - , dx y x f = < < < < - 2 1 6 1 0 6 2 0 2 0 2 y dx y x y dx y x y y = ( 29 ( 29 < < < < = - = = = 2 1 2 1 0 2 0 2 3 0 3 y y x y y x x y x x y x = ( 29 < < - < < 2 1 2 2 1 0 2 3 4 y y y y y
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2. Suppose that ( X, Y ) is uniformly distributed over the region defined by – 1 x 1 and 0 y 1 – x 2 . a)
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Hw09ans - STAT 408 Spring 2009 Homework#9(due Friday April...

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