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 a mean and variance of \(n\) and \(2n\), respectively.

- The total area under a \(\chi^2\)-curve equals 1.
- A \(\chi^2\)-curve starts at 0 on the horizontal axis and extends indefinitely to the right, approaching, but never touching, the horizontal axis as it does so.
- A \(\chi^2\)-curve is right
skewed.

- As the number of degrees of freedom becomes larger, \(\chi^2\)- curves look increasingly like normal curves.

**Citation**

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

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Please cite as follow: *Hartmann,
K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis
using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.*