Chi distribution

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. Equivalently, it is the distribution of the Euclidean distance between a multivariate Gaussian random variable and the origin. It is thus related to the chi-squared distribution by describing the distribution of the positive square roots of a variable obeying a chi-squared distribution.

chi

Probability density function
Cumulative distribution function
Parameters	${\displaystyle k>0\,}$ (degrees of freedom)
Support	${\displaystyle x\in [0,\infty )}$
PDF	${\displaystyle {\frac {1}{2^{(k/2)-1}\Gamma (k/2)}}\;x^{k-1}e^{-x^{2}/2}}$
CDF	${\displaystyle P(k/2,x^{2}/2)\,}$
Mean	${\displaystyle \mu ={\sqrt {2}}\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}}$
Median	${\displaystyle \approx {\sqrt {k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}}}}$
Mode	${\displaystyle {\sqrt {k-1}}\,}$ for ${\displaystyle k\geq 1}$
Variance	${\displaystyle \sigma ^{2}=k-\mu ^{2}\,}$
Skewness	${\displaystyle \gamma _{1}={\frac {\mu }{\sigma ^{3}}}\,(1-2\sigma ^{2})}$
Ex. kurtosis	${\displaystyle {\frac {2}{\sigma ^{2}}}(1-\mu \sigma \gamma _{1}-\sigma ^{2})}$
Entropy	${\displaystyle \ln(\Gamma (k/2))+\,}$ ${\displaystyle {\frac {1}{2}}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi _{0}(k/2))}$
MGF	Complicated (see text)
CF	Complicated (see text)

If ${\displaystyle Z_{1},\ldots ,Z_{k}}$ are ${\displaystyle k}$ independent, normally distributed random variables with mean 0 and standard deviation 1, then the statistic

${\displaystyle Y={\sqrt {\sum _{i=1}^{k}Z_{i}^{2}}}}$

is distributed according to the chi distribution. The chi distribution has one positive integer parameter ${\displaystyle k}$, which specifies the degrees of freedom (i.e. the number of random variables ${\displaystyle Z_{i}}$).

The most familiar examples are the Rayleigh distribution (chi distribution with two degrees of freedom) and the Maxwell–Boltzmann distribution of the molecular speeds in an ideal gas (chi distribution with three degrees of freedom).