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Thread: Joint Probability Distribution

  1. #1
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    Joint Probability Distribution

    Let X and Y be two random variables with joint density function f_XY. Compute the pdf of U = XY.

    Here is my work.

    U=XY and V=X so X=V and Y=U/V

    f_U(u) = integral (-inf to inf)f_U(u)dv = integral (-inf to inf)(f_XY(v,u/v)dv

    Apparently my answer is off by a factor of 1/|v| (which just happens to be |Jacobian|). I just don't see where 1/|v| comes from.

    Please advise me on this.

    Thanks!
    Last edited by JaguarXJS; Jan 21st 2017 at 08:13 PM.
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  2. #2
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    Re: Joint Probability Distribution

    Hey JaguarXJS.

    Have you done a change of variables? A Jacobian is involved when you change the co-ordinate system [or the limits of the integral].
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  3. #3
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    Re: Joint Probability Distribution

    Oh, OK. I thought that the theorem in my book said that you need the Jacobian only when you go between f(u,v) and f(x,y). I need to reread the the theorem and examine the proof.
    I appreciate your help.
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