NotesOct20 - Computing the pdf of Y = T(X Let X be...

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Computing the pdf of Y = T ( X ) . Let X be continuous with pdf f X . Sometimes Y = T ( X ) is also continuous. To compute the density of Y adjust by the derivative of the inverse transfor- mation .
Image of page 1

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

View Full Document Right Arrow Icon
Monotone transformations : the function T is either increasing or decreasing on the range of X . A monotone function T is automatically one- to-one. The pdf of Y : f Y ( y ) = f X T - 1 ( y ) dT - 1 ( y ) dy .
Image of page 2
Example Let X be a standard exponential random variable. Compute the density of Y = 1 /X .
Image of page 3

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

View Full Document Right Arrow Icon
This method sometimes works even when the function T is not monotone. The equation T ( x ) = y may have several roots, T - 1 1 ( y ), T - 1 2 ( y ) , . . . . Each of the roots contributes to the den- sity of Y = T ( X ): f Y ( y ) = X i f X T - 1 i ( y ) dT - 1 i ( y ) dy .
Image of page 4
Example Let X be standard normal, and Y = T ( X ), with T ( x ) = ( - x if x 0 2 x if x > 0. Compute the pdf of Y .
Image of page 5

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

View Full Document Right Arrow Icon
Transformations of random vectors Statement of the problem: let ( X 1 , . . . , X k ) be a random vector with a known joint pmf (if discrete), or a known joint pdf (if continuous). Let ( Y 1 , . . . , Y k ) = T ( X 1 , . . . , X k ) be a function, or a transformation, of ( X 1 , . . . , X k ). Find the joint pmf of ( Y 1 , . . . , Y k ) in the discrete case; find the joint pdf of ( Y 1 , . . . , Y k ) in the con- tinuous case.
Image of page 6
If ( X 1 , . . . , X k ) is discrete, then ( Y 1 , . . . , Y k ) = T ( X 1 , . . . , X k ) is also discrete. The joint pmf of Y 1 , . . . , Y k ): p Y 1 ,...,Y k ( y j 1 , . . . , y j k ) = X x i 1 ,...,x
Image of page 7

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

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

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

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern