# chi-squared random variable

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 (http://planetmath.org/NormalRandomVariable) $X_{1},X_{2},\ldots,X_{n}$,

 $X=X_{1}^{2}+X_{2}^{2}+\cdots+X_{n}^{2}.$

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. 1.
2. 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. 3.

$E[X]=n$

4. 4.

$\operatorname{Var}[X]=2n$

5. 5.
 $M_{X}(t)=\left(1-2t\right)^{-\frac{n}{2}},$

and is defined for all $t\in\mathbb{C}$ with real part  (http://planetmath.org/Complex) less than $1/2$.

6. 6.

The sum of independent $\chi_{(m)}^{2}$ and $\chi_{(n)}^{2}$ random variables  has the $\chi_{(m+n)}^{2}$ distribution.

Title chi-squared random variable ChisquaredRandomVariable 2013-03-22 11:54:49 2013-03-22 11:54:49 mathcam (2727) mathcam (2727) 16 mathcam (2727) Definition msc 60-00 msc 11-00 msc 20-01 msc 20A05 central chi-squared distribution ChiSquaredStatistic