
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.
Thanks

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.

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

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.