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# 06_26 - STAT 410 Examples for Summer 2009 Let X 1 and X 2...

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STAT 410 Examples for 06/26/2009 Summer 2009 Let X 1 and X 2 have joint p.d.f. f ( x 1 , x 2 ) . S = { ( x 1 , x 2 ) : f ( x 1 , x 2 ) > 0 } – support of ( X 1 , X 2 ) . F ( x 1 , x 2 ) = P ( X 1 x 1 , X 2 x 2 ) . f ( x 1 , x 2 ) = 2 F ( x 1 , x 2 ) / x 1 x 2 . Let Y 1 = u 1 ( X 1 , X 2 ) and Y 2 = u 2 ( X 1 , X 2 ) . y 1 = u 1 ( x 1 , x 2 ) y 2 = u 2 ( x 1 , x 2 ) one-to-one transformation maps S onto T – support of ( Y 1 , Y 2 ) . x 1 = w 1 ( y 1 , y 2 ) x 2 = w 2 ( y 1 , y 2 ) J = 2 2 1 2 2 1 1 1 y x y x y x y x The joint p.d.f. g ( y 1 , y 2 ) of ( Y 1 , Y 2 ) is f ( w 1 ( y 1 , y 2 ) , w 2 ( y 1 , y 2 ) ) | J | ( y 1 , y 2 ) T g ( y 1 , y 2 ) = 0 elsewhere. 1. Let X 1 and X 2 have joint p.d.f. f ( x 1 , x 2 ) = 2 e ( x 1 + x 2 ) , 0 < x 1 < x 2 . a) Find the joint p.d.f. g ( y 1 , y 2 ) of the variables Y 1 = X 2 – X 1 and Y 2 = X 1 . b) Find the joint p.d.f. h ( z 1 , z 2 ) of the variables Z 1 = X 1 + X 2 and Z 2 = X 2 / X 1 .

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2. Let X 1 and X 2 have the joint probability density function f X 1 , X 2 ( x 1 , x 2 ) = < < < otherwise 0 1 0 15 1 2 2 2 1 x x x x a) Let Y 1 = X 1 + X 2 and Y 2 = X 1 . Find the joint probability density function of ( Y 1 , Y 2 ) , f Y 1 , Y 2 ( y 1 , y 2 ) . Sketch the support of ( Y 1 , Y 2 ) . b) Let Y 1 = X 2 / X 1 and Y 2 = X 1 + X 2 . Find the joint probability density function of ( Y 1 , Y 2 ) , f Y 1 , Y 2 ( y 1 , y 2 ) . Sketch the support of ( Y 1 , Y 2 ) . 3. 2.2.5
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• Summer '08
• AlexeiStepanov
• Probability distribution, Probability theory, probability density function, joint probability density

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