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Math Help - finding the PDF!!

  1. #1
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    Question finding the PDF!!

    f(x,y) = θe-(x+θy) , θ>0 and x>0

    find the PDF of XY

    thanks
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  2. #2
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    Quote Originally Posted by lauren2988 View Post
    f(x,y) = θe-(x+θy) , θ>0 and x>0

    find the PDF of XY

    thanks
    Let U = XY.

    Calculate the cumulative density function F(u). Then the pdf of U is g(u) = \frac{dF}{du}.

    F(u) = \Pr(U < u) = \Pr(XY < u) = \Pr\left( Y < \frac{u}{X}\right) = \int_{x = -\infty}^{+\infty} \int_{y = -\infty}^{+\infty} f(x, y) \, dy \, dx

    Are you given that the joint pdf has the exponential rule for y > 0 as well as x > 0? If not, then the next line won't be correct:

     = \int_{x=0}^{x = +\infty} \int_{y=0}^{y = u/x} \theta \, e^{-(x + \theta y)} \, dy \, dx.

    The calculus stuff is left for you to do.
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