NotesOct22

# NotesOct22 - Let(X1 Xk be continuous with a joint pdf...

• Notes
• 5

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

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 .

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

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 )) .
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 (

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

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 ) 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 )....
View Full Document

• '10
• SAMORODNITSKY
• yk, inverse transformation

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern