09_16 - x , 0 < x < , zero elsewhere....

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STAT 410 Examples for 09/16/2011 Fall 2011 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 g ( y 1 , y 2 ) = f ( w 1 ( y 1 , y 2 ) , w 2 ( y 1 , y 2 ) ) | J | ( y 1 , y 2 ) T 0 elsewhere.
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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 and Y have the joint probability density function f X, Y ( x , y ) = < + > > otherwise 0 1 , 0 , 0 60 2 y x y x y x Let U = X Y and V = X. Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ). 3. 2.7.1 Let X 1 , X 2 , X 3 be iid, each with the distribution having pdf f ( x ) = e
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Unformatted text preview: x , 0 &lt; x &lt; , zero elsewhere. Show that Y 1 = 2 1 1 X X X + , Y 2 = 3 2 1 2 1 X X X X X + + + , Y 3 = X 1 + X 2 + X 3 are mutually independent. 4. 2.2.5 Let X 1 and X 2 be continuous random variables with the joint probability density function, f X 1 , X 2 ( x 1 , x 2 ), &lt; x i &lt; , i = 1, 2. Let Y 1 = X 1 + X 2 and Y 2 = X 2 . (a) Find the joint pdf f Y 1 , Y 2 . (b) Show that ( ) ( ) --= 2 2 2 1 X , X 1 Y , 2 1 1 dy y y y f y f ( 2.2.1 ) which is sometimes called the convolution formula . 5. 2.2.7 Use the formula ( 2.2.1 ) to find the pdf of Y 1 = X 1 + X 2 , where X 1 and X 2 have the joint pdf f X 1 , X 2 ( x 1 , x 2 ) = 2 e ( x 1 + x 2 ) , 0 &lt; x 1 &lt; x 2 &lt; , zero elsewhere....
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This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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09_16 - x , 0 &amp;amp;lt; x &amp;amp;lt; , zero elsewhere....

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