The Chi-Squared distribution with degrees of freedom \(n\) (argument df) and denoted as \(\chi_{n}^2\), where \(n>0\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995)
$$f(y)= \displaystyle \frac{1}{2^{n/2}\Gamma\left(\frac{n}{2}\right)}y^{n/2-1}e^{-y/2},$$
and a cumulative distribution function given by
$$F(y) = \frac{\gamma\left(\frac{n}{2}, \frac{y}{2}\right)}{\Gamma(\alpha)},$$
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
$$G=\displaystyle \frac{2\Gamma\left( \frac{1+n}{2}\right)}{n\Gamma\left(\frac{n}{2}\right)\sqrt{\pi}}.$$ The Chi-Squared distribution is related to the Gamma distribution: \(\chi_{n}^2 = Gamma(n/2, 2)\).