Hw03ans - STAT 410 Spring 2008 Homework#3(due Friday...

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Unformatted text preview: STAT 410 Spring 2008 Homework #3 (due Friday, February 8, 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 ). b) What must the value of C be so that f ( x , y ) is a valid joint p.d.f.? Must have ( ) & & ∞ ∞ ∞ ∞-- dy dx y x f , = 1. & & ¡ ¡ ¢ £ ¤ ¤ ¥ ¦- 1 2 2 dx dy y x x x C = & =- = ¡ ¢ £ ¤ ¥ ¦ 1 2 2 2 2 dx y x x y x y C = ( ) [ ] & ¡ ¢ £ ¤ ¥ ¦-- 1 2 2 2 2 2 dx x x x C = ( ) &- 1 3 2 2 2 dx x x C C = 1 4 3 2 3 2 ¡ ¢ £ ¤ ¥ ¦- x x C C = 6 C = 1. § C = 6 . c) Find P ( X + Y < 1 ). & & ¡ ¡ ¢ £ ¤ ¤ ¥ ¦- 5 . 1 2 6 dx dy y x x x = ( ) & =- = 5 . 1 2 2 3 dx y x x y x y = ( ) [ ] ( ) &-- 5 . 2 2 2 1 3 dx x x x = ( ) &- 5 . 3 2 6 3 dx x x = 5 . 4 3 2 3 ¡ ¢ £ ¤ ¥ ¦- x x = 4 3 2 1 2 3 2 1 ¡ ¢ £ ¤ ¥ ¦- ¡ ¢ £ ¤ ¥ ¦ = 32 3 8 1- = 32 1 = 0.03125 . 2. Suppose that the random variables X and Y have joint p.d.f. f ( x , y ) given by f ( x , y ) = 6 x 2 y , 0 < x < y , x + y < 2. a) Find the marginal probability density function for X. First, X can only take values in ( , 1 ). f X ( x ) = ( ) & ∞ ∞- , dy y x f = &- x x dy y x 2 2 6 = ( ) 3 2 2 2 x y x y y x =- = = ( ) { } 2 2 2 2 3 x x x-- = 12 x 2 – 12 x 3 = 12 x 2 ( 1 – x ), 0 < x < 1. b) Find the marginal probability density function for Y. First, Y can only take values in ( , 2 ). f Y ( y ) = ( ) & ∞ ∞- , dx y x f = ¡ ¡ ¡ ¢ ¡ ¡ ¡ £ ¤ < < < < & &- 2 1 6 1 6 2 2 2 y dx y x y dx y x y y = ( ) ( ) ¡ ¡ ¡ ¢ ¡ ¡ ¡ £ ¤ < < < < =- = = = 2 1 2 1 2 2 3 3 y y x y y x x y x x y x = ( ) ¡ ¡ ¢ ¡ ¡ £ ¤ < <- < < 2 1 2 2 1 2 3 4 y y y y y From the textbook:...
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Hw03ans - STAT 410 Spring 2008 Homework#3(due Friday...

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