/Users/jokr/Documents/soga/2046_chi-squared

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 the mean and variance equal to $n$ 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.