NotesOct22 - k ( y 1 ,...,y k ) = = f X 1 ,...,X k T-1 ( y...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 )) .
Background image of page 2
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 ( Y 1 ,...,Y k ): f Y 1 ,...,Y
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: k ( y 1 ,...,y k ) = = f X 1 ,...,X k T-1 ( y 1 ,...,y k ) J T-1 ( y 1 ,...,y k ) . 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 )....
View Full Document

Page1 / 5

NotesOct22 - k ( y 1 ,...,y k ) = = f X 1 ,...,X k T-1 ( y...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online