Results 1 to 5 of 5

Math Help - transformation function of random variables

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    12

    transformation function of random variables

    If X and Y are exponentially distributed with the same parameter and they are also independent:

    To compute the density function of :

    Case 1) Z = X-Y with the constraint that X is strictly higher than Y, (X>Y)

    Case 2) Z = X+Y without any constraint

    If I want to do it by computing the cdf first:

    Can someone explain which should be the limits in the following integral in both cases:

    <br /> <br />
F_Z(z) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}f_X(x)f_Y(y) dx dy<br />


    Thanks in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    Case 2) Z = X+Y without any constraint

    is easy, you're adding two independent gammas, use the MGF technique here


    Case 1) Z = X-Y with the constraint that X is strictly higher than Y
    Here you can draw the region.
    I would get the joint density and just integrate the infinite triangle in the first quadrant
    where X>Y
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2009
    Posts
    12
    First of all thank you!

    For the first case is OK. It is just the convolution, however in the second case I do not reach the rigth solution, I get lost with the integral limits in getting the pdf of Z= X-Y...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    F_Z(a)=P(Z\le a)=P(X-Y\le a)

    NOW draw this region.
    It's in the first quadrant with the boundary x-y=a or y=x-a.
    You have the region that has vertex (a,0) in the first quadrant.

    So it's the region between x-y=a and x-y=0 (that's y=x)

    If that's true, dxdy is preferable

    {1\over \beta^2}\int_0^{\infty}\int_y^{y+a}e^{-(x+y)/\beta}dxdy
    Last edited by matheagle; September 30th 2009 at 12:24 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Sep 2009
    Posts
    12
    Thank you very much. I finally got the idea!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: December 13th 2011, 01:07 AM
  2. Function of random variables
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: June 8th 2011, 03:55 AM
  3. transformation of independent random variables
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 11th 2010, 11:23 PM
  4. Function of random variables
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: July 28th 2009, 09:26 AM
  5. cdf of a transformation of two random variables
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 14th 2008, 03:02 PM

Search Tags


/mathhelpforum @mathhelpforum