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Math Help - Just 2 questions...

  1. #16
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    Quote Originally Posted by mr fantastic View Post
    where f(x) = \frac{1}{\lambda} \, e^{-x/\lambda} ....
    Yea, but I was trying to convert from (s,t) -> to (x,y).
    By right, the answers should tally, but they don't , so I am guessing the original question was wrong.

    Shouldn't it be :

    <br />
f_{ST}(s,t) = \frac{\lambda^{-2} s}{(1+t)^2} e^{- \lambda s}<br />
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  2. #17
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    Quote Originally Posted by chopet View Post
    Yea, but I was trying to convert from (s,t) -> to (x,y).
    By right, the answers should tally, but they don't , so I am guessing the original question was wrong. Mr F asks: What original question??

    I have no idea what question you're trying to do but there's certainly nothing wrong with any of the questions asked in this thread.

    Shouldn't it be :

    <br />
f_{ST}(s,t) = \frac{\lambda^{-2} s}{(1+t)^2} e^{- \lambda s}<br />
    ..
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  3. #18
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    Quote Originally Posted by mr fantastic View Post
    ..
    I'm trying to do a transformation of variables from (s,t) -> (x,y) without using the fact that f(x) and f(y) are exponential distributions.
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  4. #19
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    Quote Originally Posted by chopet View Post
    I'm trying to do a transformation of variables from (s,t) -> (x,y) without using the fact that f(x) and f(y) are exponential distributions.
    S = X + Y .... (1)
    T = X/Y .... (2)

    f_{S, T} (s,t) = \frac{\lambda^2 s}{(1 + t)^2} \, e^{-\lambda \, s}.

    From (1) and (2):

    X = ST/(1 + T)

    Y = S/(1 + T)

    |J| = \frac{s}{(1 + t)^2}.

    Therefore f_{X,Y} (x,y) = f_{S, T} (s,t)/|J| = \lambda^2 e^{-\lambda s} = \lambda^2 e^{-\lambda (x +  y)}.

    This distribution is obviously equivalent to f_{X,Y} (x,y) = \frac{1}{\lambda^2} e^{-(x +  y)/\lambda} and there's no problem that I can see.


    Edit: I see that you get it now. Excellent. Now the trick is to derive the given joint distribution for S and T from the distributions for X and Y lol!
    Last edited by mr fantastic; September 3rd 2008 at 02:00 PM. Reason: Fixed a small bit of latex
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