20460_chi-squared_distribution.knit

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

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

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.

Basic Properties of $\chi^2$-curves (Weiss 2010):

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.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

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.

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