Probability Density Function
The parameter is \(\nu\) and it is a positive integer.
It is the gamma distribution with \(\alpha=\frac{1}{2},\beta=2\)
\begin{equation*}
f(x;\nu)=\frac{x^{\left(\frac{\nu}{2}\right)-1}e^{-\frac{x}{2}}}{2^{\frac{\nu}{2}}\Gamma\left(\frac{\nu}{2}\right)}
\end{equation*}
For \(x\ge0\)
\(\nu\) is called the number of degrees of freedom (df) of \(X\).
Relation to the Normal Distribution
It is the distribution of a sum of the squares of \(\nu\) independent standard normal random variables.
Usage
It is rarely used to model natural phenomena. However, it is used heavily in inferential statistics.