Given an natural number r, suppose X is a random variable over the space \(R = (0,\infty)\) with probability function given by
\begin{equation*}
f(x) = \frac{x^{r/2-1} e^{-x/2} }{\Gamma(r/2) 2^{r/2}}.
\end{equation*}
Then X has a Chi-Square distribution with r degrees of freedom. This is often denoted \(\chi^2(r)\text{.}\)