A central chi-squared random variable $X$ with $n>0$ degrees of freedom is given by the probability density function $$ f_X(x) = \frac{ (\frac{1}{2})^{\frac{n}{2}} } {\Gamma(\frac{n}{2})} x^{\frac{n}{2} - 1} e^{- \frac{1}{2} x} $$ for $x > 0$ , where $\Gamma$ represents the gamma function.

The parameter $n$ is usually, but not always, an integer, in which case the distribution is that of the sum of the squares of a sequence of $n$ independent standard normal variables $X_1,X_2,\ldots,X_n$ , \begin{equation*} X=X_1^2+X_2^2+\cdots+X_n^2. \end{equation*} Parameters: $n\in(0,\infty)$ .

Syntax: $X\sim \chi_{(n)}^{2}$

Figure 1: Densities of the chi-squared distribution for different degrees of freedom.

Notes:

1. This distribution is very widely used in statistics, such as in hypothesis tests and confidence intervals.
2. The chi-squared distribution with $n$ degrees of freedom is a result of evaluating the gamma distribution with $\alpha = \frac{n}{2}$ and $\lambda = \frac{1}{2}$ .
3. $E[X] = n$
4. $\Var[X] = 2n$
5. The moment generating function is \begin{equation*} M_X(t) = \left(1 - 2t\right)^{-\frac{n}{2}}, \end{equation*}and is defined for all $t\in\mathbb{C}$ with real part less than $1/2$ .
6. The sum of independent $\chi_{(m)}^2$ and $\chi_{(n)}^2$ random variables has the $\chi_{(m+n)}^2$ distribution.