410Hw04ans - STAT 410 Summer 2009 Homework #4 (due Monday,...

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STAT 410 Summer 2009 Homework #4 (due Monday, July 6, by 4:00 p.m.) 1. Consider two continuous random variables X and Y with joint probability density function f ( x , y ) = 6 ( 1 – x ), 0 < y < x < 1. a) Find f X | Y ( x | y ). f Y ( y ) = ( 29 - 1 1 6 y dx x = 3 – 6 y + 3 y 2 = 3 ( 1 – y ) 2 , 0 < y < 1. f X | Y ( x | y ) = ( 29 ( 29 2 1 1 2 y x - - , y < x < 1, 0 < y < 1. b) Find = 2 1 Y 4 3 X P . f X | Y ( x | 2 1 ) = 8 ( 1 – x ), 2 1 < x < 1. = 2 1 Y 4 3 X P = ( 29 - 1 4 3 1 8 dx x = 4 1 . c) Find = 2 1 Y X E . f X | Y ( x | 2 1 ) = 8 ( 1 – x ), 2 1 < x < 1. = 2 1 Y X E = ( 29 - 1 2 1 1 8 dx x x = 3 2 .
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2. Once a car accident is reported to an insurance company, the company makes an initial estimate, X, of the amount it will pay to the claimant. When the claim is finally settled, the company pays an amount, Y, to the claimant. The company has determined that X and Y have the joint p.d.f. f ( x , y ) = ( 29 ( 29 ( 29 1 1 2 2 1 2 - - - - x x y x x , x > 1, y > 1. a) Given that the initial claim estimated by the company is 1.5, determine the probability that the final settlement amount exceeds 2. f X ( x ) = ( 29 ( 29 ( 29 - - - - 1 1 1 2 2 1 2 dy y x x x x = 3 2 x , x > 1. f Y | X ( y | x ) = ( 29 ( 29 1 1 2 1 - - - - x x y x x , y > 1. f Y | X ( y | x = 1.5 ) = 4 3 - y , y > 1. P ( Y > 2 | X = 1.5 ) = - 2 4 3 dy y = 8 1 = 0.125 . b) Find E ( Y | X = x ). E ( Y | X = x ) = ( 29 ( 29 - - - - 1 1 1 2 1 dy y x x y x x = x , x > 1.
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3. Let X denote the number of times a certain numerical control machine will malfunction: 0, 1, or 2 times, on any given day. Let Y denote the number of times a technician is called on an emergency call. The joint p.m.f. p X, Y ( x , y ) is presented in the table below: X Y 0 1 2 0 0.10 0.05 0.05 1 0.10 0.15 0.15 2 0 0.15 0.25 a) Find P ( Y > X ). P ( Y > X ) = p ( 1, 0 ) + p ( 2, 0 ) + p ( 2, 1 ) = 0.25 . b) Find p X ( x ), the marginal p.m.f. for the number of machine malfunctions. p X ( x ) = = = = 2 1 0 , 45 . 0 , 35 . 0 , 20 . 0 x x x c) Find p Y | X ( y | 2 ), the conditional p.m.f. for the number of emergency calls given two machine malfunctions. p Y | X ( y | 2 ) = ( 29 ( 29 2 , 2 X p y p = ( 29 45 . 0 , 2 y p = = = = 2 1 0 , 45 . 0 25
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410Hw04ans - STAT 410 Summer 2009 Homework #4 (due Monday,...

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