The chi-square (\(\chi^2\)) distribution is one of the most important continuous probability distributions with many uses in statistical theory and inference (Lovric 2011).

Let \(n > 0\) be a positive integer. For a random variable that has a (\(\chi^2\))-distribution with \(n\) degrees of freedom \((df)\) the probability density function is

\[ f(x) = \begin{cases} 0 & \text{if $x \le 0$} \\ \frac{x^{(n/2-1)}e^{-x/2}}{2^{n/2}\Gamma \left(\frac{k}{2}\right)} & \text{if $x > 0$} \end{cases} \]

where \(\Gamma\) denotes the Gamma function. The (\(\chi^2\))-distribution (with \(n\) degrees of freedom) is equal to the \(\Gamma\)-distribution with the parameters (\(n/2,2\)), that is, with the mean and variance equal to \(n\) and \(2n\) respectively.

Basic Properties of \(\chi^2\)-Curves (Weiss 2010)