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Thread: 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:

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

    What does the subscript $\displaystyle \frac{x}{\beta}$ 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.

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

    There is no closed form expression for this guy as a function of $\displaystyle \alpha$. If you fix $\displaystyle \alpha$ and try to write as a function of $\displaystyle x$ you can sometimes do that, particularly if $\displaystyle \alpha$ is a positive integer.
    Last edited by Guy; Jun 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 $\displaystyle \alpha$ you may interchange integration and differentiation. So

    $\displaystyle \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 $\displaystyle \beta$ you can use (chain rule + fundamental theorem of calculus). You can't get rid of the integrals though.
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