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410Hw04 - STAT 410 Fall 2011 Homework#4(due Friday...

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STAT 410 Fall 2011 Homework #4 (due Friday, September 23, by 3:00 p.m.) 1. Let X be a Uniform ( 0, 1 ) and Y be a Uniform ( 0, 3 ) independent random variables. Let W = X + Y. Find and sketch the p.d.f. of W. 2. Let random variables X and Y have an Exponential distribution with mean 1 and a Uniform distribution on ( 0 , 1 ) , respectively. Suppose X and Y are independent. Find the probability density function of W = X + Y. 3. Let X and Y have the pdf f ( x , y ) = 1, 0 < x < 1, 0 < y < 1, zero elsewhere. Find the c.d.f. and the p.d.f. of W = X – Y. “Hint”: Consider two cases: 1 < w < 0 and 0 < w < 1. 4. Let X and Y have the pdf f ( x , y ) = 1, 0 < x < 1, 0 < y < 1, zero elsewhere. Find the c.d.f. and the p.d.f. of V = X / Y . “Hint”: Consider two cases: 0 < v < 1 and v > 1. 5. 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. 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 ) . 6. Suppose the joint probability density function of ( X , Y ) is f X Y ( x , y ) = otherwise 0 1 0 15 2 x y y x Let U = X + Y
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