The Gamma distribution with shape parameter \(\alpha\), scale parameter \(\sigma\) and denoted as \(Gamma(\alpha, \sigma)\), where \(\alpha>0\) and \(\sigma>0\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995)
$$f(y) = \displaystyle \frac{1}{\sigma^{\alpha}\Gamma(\alpha)}y^{\alpha-1}e^{-y/\sigma},$$
and a cumulative distribution function given by
$$F(y) = \frac{\gamma\left(\alpha, \frac{y}{\sigma}\right)}{\Gamma(\alpha)},$$
where \(y \geq 0\), the gamma function is defined by
$$\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt,$$
and the lower incomplete gamma function is given by
$$\gamma(\alpha,y) = \int_{0}^{y}t^{\alpha-1}e^{-t}dt.$$
The Gini index can be computed as
$$G = \displaystyle \frac{\Gamma\left(\frac{2\alpha+1}{2}\right)}{\alpha\Gamma(\alpha)\sqrt{\pi}}.$$ The Gamma distribution is related to the Chi-squared distribution: \(Gamma(n/2, 2) = \chi_{n}^2\).