NotesOct22 - k y 1,y k = = f X 1,X k ± T-1 y 1,y k ² ³...

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Let ( X 1 , . . . , X k ) be continuous with a joint pdf f X 1 ,...,X k ( x 1 , . . . , x k ). Sometimes ( Y 1 , . . . , Y k ) = T ( X 1 , . . . , X k ) is also continuous. Instead of adjusting by the derivative of the inverse transformation, adjust by the Jacobian of the inverse transformation .
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Suppose that the function T is one-to-one on the range of ( X 1 , . . . , X k ). The notation for the inverse transformation: ( x 1 , . . . , x k ) = T - 1 ( y 1 , . . . , y k ) = ( h 1 ( y 1 , . . . , y k ) , . . . , h k ( y 1 , . . . , y k )) .
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Compute the Jacobian of the inverse trans- formation: J T - 1 ( y 1 , . . . , y k ) = ∂h 1 ∂y 1 ∂h 1 ∂y 2 . . . ∂h 1 ∂y k ∂h 2 ∂y 1 ∂h 2 ∂y 2 . . . ∂h 2 ∂y k . . . . . . . . . . . . ∂h k ∂y 1 ∂h k ∂y 2 . . . ∂h k ∂y k The joint density of (
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Unformatted text preview: k ( y 1 ,...,y k ) = = f X 1 ,...,X k ± T-1 ( y 1 ,...,y k ) ² ³ ³ ³ J T-1 ( y 1 ,...,y k ) ³ ³ ³ . Example : Let X 1 and X 2 be independent standard exponential random variables, and let Y 1 = X 1 + X 2 , Y 2 = X 1 /X 2 . Find the joint pdf of Y 1 and Y 2 . The range of Y 1 ,Y 2 might be the issue. Example Let ( X 1 ,X 2 ) be continuous with a joint pdf f X 1 ,X 2 ( x 1 ,x 2 ) = ( 1 if 0 < x 1 ,x 2 < 1 0 otherwise. Let Y 1 = X 1 + X 2 , Y 2 = X 1-X 2 . Find the joint pdf of ( Y 1 ,Y 2 )....
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