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09_30ans - STAT 410 Examples for Fall 2009 1 Let X and Y...

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STAT 410 Examples for 09/30/2009 Fall 2009 1. Let X and Y have the joint p.d.f. f X Y ( x , y ) = 20 x 2 y 3 , 0 < x < 1, 0 < y < x . a) Find f X ( x ), f Y ( y ). f X ( x ) = 5 x 4 , 0 < x < 1. f Y ( y ) = ( 29 9 3 3 20 y y - , 0 < y < 1. b) Find f X | Y ( x | y ), f Y | X ( y | x ). f X | Y ( x | y ) = 6 2 1 3 y x - , y 2 < x < 1. f Y | X ( y | x ) = 2 3 4 x y , 0 < y < x . c) Find E ( X | Y = y ), E ( Y | X = x ). E ( X | Y = y ) = 6 8 1 1 4 3 y y - - , 0 < y < 1. E ( Y | X = x ) = x 5 4 , 0 < x < 1. d) Find E ( X ), E ( Y ). E ( X ) = 6 5 . E ( Y ) = 11 8 .

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e) Let U = Y 2 and V = X Y. Find the joint probability density function of ( U, V ), f U V ( u , v ). Sketch the support of ( U, V ). Y = U X = U V 0 < y y < x u 3 / 2 < v , x < 1 v < u . J = 0 2 1 1 2 2 1 2 1 2 3 u u u v - = u 2 1 - . | J | = u 2 1 . f U V ( u , v ) = u u u v 2 1 20 2 3 2 × = u v 2 10 , 0 < u < 1, 2 3 u < v < 2 1 u .
2. Let λ > 0. Consider the following joint probability distribution

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09_30ans - STAT 410 Examples for Fall 2009 1 Let X and Y...

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