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Math Help - Monte Carlo Method

  1. #1
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    Monte Carlo Method

    Determine a method to generate random observations for the extreme valued pdf given by:

    e^{x-e^x} \ \ \ \ \ - \infty < x < \infty

    So I start by finding it's CDF:

    \int^{x}_{-\infty}e^{x-e^x} dx

    and this is where I get stuck. Any help would be appreciated.
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  2. #2
    Master Of Puppets
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    \int e^{x-e^x} dx= -e^{-e^x}
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by statmajor View Post
    Determine a method to generate random observations for the extreme valued pdf given by:

    e^{x-e^x} \ \ \ \ \ - \infty < x < \infty

    So I start by finding it's CDF:

    \int^{x}_{-\infty}e^{x-e^x} dx

    and this is where I get stuck. Any help would be appreciated.
    It is advisable not to use the same variable name for the dummy variable of integration and the limit of integration. Instead write this as:

    \int^{x}_{-\infty}e^{\xi-e^{\xi}} d\xi

    (or use whatever name you want for the dummy variable but not x)

    CB
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  4. #4
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    \int^{x}_{-\infty}e^{t-e^t} dt = -e^{-e^t} |^x_{-\infty} = 1 -e^{-e^x}

    y = 1 -e^{-e^x} => 1 - y = e^{-e^x}
     ln(1 -y) = -e^x => -ln(1 -y) = e^x => ln(-ln(1 -y)) = x

    U ~ Uniform(0,1)

    ln(-ln(1 - U)) = F^{-1}(u) = X

    Is this correct?
    Last edited by statmajor; December 2nd 2009 at 10:11 AM.
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