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# Quiz2Aans - 15 16 5 2 1 4 15 2 5 4 3 = =-Âµ Â Â Â¸ Â¹ Âº-y...

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STAT 410 Spring 2008 Version A Name ANSWERS . Quiz 2 (10 points) Be sure to show all your work, your partial credit might depend on it. No credit will be given without supporting work. 1. Let X and Y have the joint probability density function f ( x , y ) = ± ² ³ < < < otherwise 0 1 0 15 2 x y y x a) (4) Find P ( X + Y < 1 ). P ( X + Y < 1 ) = ´ ´ µ µ µ · ¸ ¸ ¸ ¹ º - 2 1 0 1 2 15 dy dx y x y y = ´ ´ µ µ µ · ¸ ¸ ¸ ¹ º - 2 1 0 1 2 2 2 15 dy dx x y y y = ( ) ( ) ´ - - 2 1 0 2 2 2 1 2 15 dy y y y = ( ) ´ - 2 1 0 2 2 1 2 15 dy y y = ´ µ · ¸ ¹ º - 2 1 0 3 2 15 2 15 dy y y = 64 5 64
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Unformatted text preview: 15 16 5 2 1 4 15 2 5 4 3 = =-Âµ Â¶ Â· Â¸ Â¹ Âº-y y = 0.078125. b) (6) Find E ( Y | X = x ). f X ( x ) = Â´ x dy y x 2 15 = Â´ x dy y x 2 3 5 = ( ) 5 3 x y x = 5 x 4 , 0 < x < 1. f Y | X ( y | x ) = ( ) ( ) x f y x f , X = 4 2 5 15 x y x = 3 2 3 x y , 0 < y < x . E ( Y | X = x ) = Â´ â‹… x dy x y y 3 2 3 = Â´ â‹… x dy y x 3 3 3 = 4 1 3 4 3 x y x Âµ Â¶ Â· Â¸ Â¹ Âº â‹… = x â‹… 4 3 , 0 < x < 1....
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