Chi distribution

The probability density function (pdf) of the chi-distribution is

${\displaystyle f(x;k)={\begin{cases}{\dfrac {x^{k-1}e^{-x^{2}/2}}{2^{k/2-1}\Gamma \left({\frac {k}{2}}\right)}},&x\geq 0;\\0,&{\text{otherwise}}.\end{cases}}}$

where ${\displaystyle \Gamma (z)}$ is the gamma function.

The cumulative distribution function is given by:

${\displaystyle F(x;k)=P(k/2,x^{2}/2)\,}$

where ${\displaystyle P(k,x)}$ is the regularized gamma function.

The moment-generating function is given by:

${\displaystyle M(t)=M\left({\frac {k}{2}},{\frac {1}{2}},{\frac {t^{2}}{2}}\right)+t{\sqrt {2}}\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}M\left({\frac {k+1}{2}},{\frac {3}{2}},{\frac {t^{2}}{2}}\right),}$

where ${\displaystyle M(a,b,z)}$ is Kummer's confluent hypergeometric function. The characteristic function is given by:

${\displaystyle \varphi (t;k)=M\left({\frac {k}{2}},{\frac {1}{2}},{\frac {-t^{2}}{2}}\right)+it{\sqrt {2}}\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}M\left({\frac {k+1}{2}},{\frac {3}{2}},{\frac {-t^{2}}{2}}\right).}$

The raw moments are then given by:

${\displaystyle \mu _{j}=2^{j/2}{\frac {\Gamma ((k+j)/2)}{\Gamma (k/2)}}}$

where ${\displaystyle \Gamma (z)}$ is the gamma function. The first few raw moments are:

${\displaystyle \mu _{1}={\sqrt {2}}\,\,{\frac {\Gamma ((k\!+\!1)/2)}{\Gamma (k/2)}}}$

${\displaystyle \mu _{2}=k\,}$

${\displaystyle \mu _{3}=2{\sqrt {2}}\,\,{\frac {\Gamma ((k\!+\!3)/2)}{\Gamma (k/2)}}=(k+1)\mu _{1}}$

${\displaystyle \mu _{4}=(k)(k+2)\,}$

${\displaystyle \mu _{5}=4{\sqrt {2}}\,\,{\frac {\Gamma ((k\!+\!5)/2)}{\Gamma (k/2)}}=(k+1)(k+3)\mu _{1}}$

${\displaystyle \mu _{6}=(k)(k+2)(k+4)\,}$

where the rightmost expressions are derived using the recurrence relationship for the gamma function:

${\displaystyle \Gamma (x+1)=x\Gamma (x)\,}$

From these expressions we may derive the following relationships:

Mean: ${\displaystyle \mu ={\sqrt {2}}\,\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}}$

Variance: ${\displaystyle \sigma ^{2}=k-\mu ^{2}\,}$

Skewness: ${\displaystyle \gamma _{1}={\frac {\mu }{\sigma ^{3}}}\,(1-2\sigma ^{2})}$

Kurtosis excess: ${\displaystyle \gamma _{2}={\frac {2}{\sigma ^{2}}}(1-\mu \sigma \gamma _{1}-\sigma ^{2})}$

The entropy is given by:

${\displaystyle S=\ln(\Gamma (k/2))+{\frac {1}{2}}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi ^{0}(k/2))}$

where ${\displaystyle \psi ^{0}(z)}$ is the polygamma function.