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Math Help - Transformations

  1. #1
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    Transformations

    Hello,
    A question from my homework asks if u,v,w are iid exponentially and y=u/w and z=v/w find the pdf of (y,z). I know generally how to do transformations but in this case I cant find a Jacobian since the transformation is from 3 to 2 variables. How would I go about solving this problem.
    Thanks
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  2. #2
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    Hello,

    You're correct, a transformation has to be from n to n variables. (it has to be a \mathcal{C}^1-diffeomorphism)
    Thus just make the following transformation : (u,v,w) -> (y,z,w), then integrate with respect to w to get the "marginal" pdf of (y,z)
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  3. #3
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    Thank you,
    I made the necessary substitutions and found that the inverse Jacobian is W^2. I also found that (u,v,w)= (x*w,y*w,w). Where do I go from there to find f(x,y,w).
    Thank you
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  4. #4
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    Change of Variable -- the Jacobian

    what's preventing you from continuing ?
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