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  • Transformation Problem?

    I don't have a clue how to work this one out. The solution I've seen doesn't make sense to me... Can anyone explain this in detail?

    X & Y have a joint distribution with pdf f(x,y) = exp^-(x+y), x>0 , y>0.

    The random variable Z is defined to be equal to U = exp^-(x+y).

    Find the pdf of U, f(u).

    Answer is f(u) = -ln u for 0<u<1

    Thanks for the help!

  • #2
    Transformation Problem?

    Originally posted by AlG View Post
    I don't have a clue how to work this one out. The solution I've seen doesn't make sense to me... Can anyone explain this in detail?

    X & Y have a joint distribution with pdf f(x,y) = exp^-(x+y), x>0 , y>0.

    The random variable Z is defined to be equal to U = exp^-(x+y).

    Find the pdf of U, f(u).

    Answer is f(u) = -ln u for 0<u<1

    Thanks for the help!
    Hi everyone. This is my first post, and I'll try to take a crack at this one.

    Define U=e^-(X+Y) and W=Y
    Substituting W for Y we have:
    U=e^-(X+W)

    Solve for X:
    ln(U)=-X -W
    X=-W -ln(U)

    And from above:
    Y=W

    Now we can do the transformation:

    f(u,w)=f(x(u,w),y(u,w))|dxdy/dudw|
    = e^-(-W -ln(U) + W)* jacobian(|dxdy/dudw|)
    =e^-(-ln(U)*|((-U^-1*1)-(-1*0))|
    =e^(ln(U))*U^-1
    =U*U^-1
    =1

    So, f(u,w) = 1.

    To get the marginal density function of U, we need to find the domain for W to properly set up the integral.

    0<x<inf implies 0<-w-ln(u)<inf, so we have
    w<-ln(u)<inf

    0<y<inf implies 0<w<inf, so we combine from above and then we have
    0<w<-ln(u), which give the domain for w.

    f(u)= int (f(u,w) dw) over domain of w = int (dw) from 0 to -ln(u) =-ln(u)

    Since U=e^-(x+y), and x and y are both positive, U must be between 0 and 1.

    I hope this helps you.
    Jim

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    • #3
      Thanks for the reply....

      I need to review transformations, I wouldn't have known how to start off the problem.

      Have you seen a similar problem to this, or were you able to work this out based on your knowledge of
      how to work with transformations? In other words this seems like a "been there done that" problem. It'd be one I'd skip unless I've seen it before.

      Comment


      • #4
        Originally posted by AlG View Post
        Thanks for the reply....

        I need to review transformations, I wouldn't have known how to start off the problem.

        Have you seen a similar problem to this, or were you able to work this out based on your knowledge of
        how to work with transformations? In other words this seems like a "been there done that" problem. It'd be one I'd skip unless I've seen it before.
        I've done some transformation problems before, but I've never seen a problem where a random variable (u in this case) happened to be the same as the pdf of x,y. I think that's the most confusing element in this problem.

        I just stuck to what I knew and applied it to this problem: get x and y in terms of u and w, determine f(u,w), determine domain/range of u and w, and finally find the marginal density function of u.

        Comment


        • #5
          I think it would help if you understood WHAT a Jacobian is/does and why the transformation integral is set up the way it is. Do you?

          Comment

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