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Math Help - Gamma CDF

  1. #1
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    Gamma CDF

    Hello,

    How does one interpret the subscript on a gamma function? For example, in the two parameter Gamma Cumulative Distribution Function:

    F(x)=\frac{\Gamma_{\frac{x}{\beta}}(\alpha)} {\Gamma {(\alpha)} }

    What does the subscript \frac{x}{\beta} mean in terms of how I should evaluate the gamma function.


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  2. #2
    Guy
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    Re: Gamma CDF

    I believe that denotes the incomplete gamma function.

    \Gamma_{\frac x \beta} (\alpha) = \int_0 ^ {\frac x \beta} t^{\alpha - 1} e ^ {-t} \ dt.

    There is no closed form expression for this guy as a function of \alpha. If you fix \alpha and try to write as a function of x you can sometimes do that, particularly if \alpha is a positive integer.
    Last edited by Guy; June 25th 2011 at 09:21 PM.
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  3. #3
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    Re: Gamma CDF

    Any advice as to how I might take the derivative of this w.r.t. alpha and beta?
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  4. #4
    Guy
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    Re: Gamma CDF

    You can but it probably will not be so useful for whatever you are doing. To take derivative wrt \alpha you may interchange integration and differentiation. So

    \displaystyle \frac d {d\alpha} \int_0 ^ {x/\beta} t^{\alpha - 1} e^{-t} \ dx = \int_0 ^ {x/\beta} \log(t) t^{\alpha - 1} e^{-t} \ dt .

    To take derivative wrt \beta you can use (chain rule + fundamental theorem of calculus). You can't get rid of the integrals though.
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