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# 410Hw05 - STAT 410 Spring 2013 Homework#5(due Friday March...

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STAT 410 Spring 2013 Homework #5 (due Friday, March 1, by 3:00 p.m.) 1. Let the joint probability density function for ( X , Y ) be f ( x , y ) = 3 y x , 0 < x < 2, 0 < y < 1, zero otherwise. Let U = X + Y and V = X Y + 1. Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ). 2. Let X and Y have the joint probability density function f X , Y ( x , y ) = x 1 , x > 1, 0 < y < x 1 , zero elsewhere. Let U = Y and V = Y / X. Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ). 3. Let > 0. Consider two continuous random variables X and Y with joint p.d.f. f X, Y ( x , y ) = x e y 2 θ 2 θ , x > 0, 0 < y < x 2 . Let U = X and V = X / Y . Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ). 4. 2.7.4 Let X 1 , X 2 , X 3 be iid with common pdf f ( x ) = e x , x > 0, zero elsewhere. Find the joint pdf of Y 1 = X 1 , Y 2 = X 1 + X 2 , Y 3 = X 1 + X 2 + X 3 .

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5. A company provides earthquake insurance. The premium X is modeled by the p.d.f. f X ( x ) = 5 2 5 x e x , 0 < x < , while the claims Y have the p.d.f.
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410Hw05 - STAT 410 Spring 2013 Homework#5(due Friday March...

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