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Homework Help: Transform f(U,V) to f(X,Y)

  1. Sep 1, 2006 #1
    given
    [tex] f(U,V) = \lambda^{2} e^{-(u+v)\lambda}[/tex]

    How do I get:
    f(X,Y)
    where
    X = U+V
    Y =UV

    all I'm able to get is
    [tex] f(X,Y) = \lambda^{2} \e^{-(x)\lambda} |J|[/tex]

    where J is the Jacobian.
    But the Jacobian is too complicated since I was able to solve that:
    [tex]U = \frac{X + \sqrt{X^{2} - 4Y}}{2}[/tex]
    and
    [tex]V = \frac{2Y}{X + \sqrt{X^{2} - 4Y}}[/tex]

    help please.
     
  2. jcsd
  3. Sep 1, 2006 #2

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    Why do you need the Jacobian? You're not finding df, or anything like that.
     
  4. Sep 2, 2006 #3
    I neglect to mention above that f(U,V) and f(X,Y) are density functions

    that is
    [tex]f(U,V) = \lambda^{2} e^{-(u+v)\lambda}[/tex]
    for
    [tex]u \geq 0, v\geq 0[/tex]

    isn't the Jacobian needed when you are transforming variables for a distribution?
     
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