Pr1 - STAT 410 Chapter 2 Practice Problems Spring 2008 1 a...

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STAT 410 Chapter 2 Practice Problems Spring 2008 1. a) Consider a continuous random variable X with the p.d.f. f X ( x ) = e x , x > 0. Let Y = X 2 . Find the p.d.f. of Y, f Y ( y ) . b) Consider a continuous random variable X with the p.d.f. f X ( x ) = 4 24 x , x > 2. Let Y = X 1 . Find the p.d.f. of Y, f Y ( y ) . 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 . a) Find f X ( x ) , f Y ( y ) , f X | Y ( x | y ) , f Y | X ( y | x ) , E ( X ) , E ( Y ) , E ( X | Y = y ) , E ( Y | X = x ) . b) 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 ) . 3. Let X and Y have the joint probability density function f X Y ( x , y ) = x , x > 0, 0 < y < e x , zero elsewhere. a) Find f X ( x ) and f Y ( y ) . b) Find f X | Y ( x | y ) and f Y | X ( y | x ) . c) Find E ( X | Y = y ) and E ( Y | X = x ) . d) Find E ( X ) and E ( Y ) . e) Are X and Y independent?
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4. 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.
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