Gamma CDF

**rainer** · June 25th 2011, 07:36 AM

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.

Thanks

**Guy** · June 25th 2011, 07:50 AM

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.

**rainer** · June 26th 2011, 07:46 AM

Any advice as to how I might take the derivative of this w.r.t. alpha and beta?

**Guy** · June 26th 2011, 08:01 AM

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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