The Chi-Squared Distribution

Posted by Beetle B. on Tue 06 June 2017

Chi-squared distribution

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.


It is rarely used to model natural phenomena. However, it is used heavily in inferential statistics.