chi-squared random variable

A central chi-squared random variable $X$ with $n>0$ degrees of freedom is given by the probability density function

$$f_X(x)=\frac{{(\frac{1}{2})}^{\frac{n}{2}}}{\mathrm{\Gamma }(\frac{n}{2})}{x}^{\frac{n}{2}-1}{e}^{-\frac{1}{2}x}$$

for $x>0$, where $\mathrm{\Gamma }$ 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) $X_1,X_2,\mathrm{\dots },X_n$,

$$X=X_1^2+X_2^2+\mathrm{\cdots }+X_n^2.$$

Parameters: $n\in (0,\mathrm{\infty })$.

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. $\mathrm{Var}[X]=2n$

5. The moment generating function is

   $$M_X(t)={\left(1-2t\right)}^{-\frac{n}{2}},$$

   and is defined for all $t\in ℂ$ with real part (http://planetmath.org/Complex) 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.