410Hw02 - STAT 410 Fall 2011 Homework #2 (due Friday,...

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Fall 2011 Homework #2 (due Friday, September 9, by 3:00 p.m.) 1. Let X and Y have the joint p.d.f. f X Y ( x , y ) = C x 2 y 3 , 0 < x < 1, 0 < y < x , zero elsewhere. a) What must the value of C be so that f X Y ( x , y ) is a valid joint p.d.f.? b) Find P ( X + Y < 1 ). c) Let 0 < a < 1. Find P ( Y < a X ). d) Let 0 < a < 1. Find P ( X Y < a ). 2. 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 , zero elsewhere. a) Find f X ( x ). b) Find E ( X ). c) Find f Y ( y ). d) Find E ( Y ). e) Find Cov ( X, Y ). 3. Suppose the joint probability density function of ( X , Y ) is ( ) = otherwise 0 1 0 , 2 x y y x C y x f a) Find the value of C that would make ( ) y x f , a valid probability density function. b) Find P ( X > 2 Y ) . c) Find P ( X + Y < 1 ). d) Find the marginal densities of X and Y. Are X and Y independent? e)
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410Hw02 - STAT 410 Fall 2011 Homework #2 (due Friday,...

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