Chi distribution

Properties

Moments

The raw moments are then given by:

${\displaystyle {\mu }_j={\int }_0^{\mathrm{\infty }}f(x;k)x^j\mathrm{d}x=2^{j/2}\frac{\mathrm{\Gamma }\left(\frac{1}{2}(k+j)\right)}{\mathrm{\Gamma }\left(\frac{1}{2}k\right)}}$

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

${\displaystyle {\mu }_1=\sqrt{2}\frac{\mathrm{\Gamma }\left(\frac{1}{2}(k+1)\right)}{\mathrm{\Gamma }\left(\frac{1}{2}k\right)}}$

${\displaystyle {\mu }_2=k,}$

${\displaystyle {\mu }_3=2\sqrt{2}\frac{\mathrm{\Gamma }\left(\frac{1}{2}(k+3)\right)}{\mathrm{\Gamma }\left(\frac{1}{2}k\right)}=(k+1){\mu }_1,}$

${\displaystyle {\mu }_4=(k)(k+2),}$

${\displaystyle {\mu }_5=4\sqrt{2}\frac{\mathrm{\Gamma }\left(\frac{1}{2}(k+5)\right)}{\mathrm{\Gamma }\left(\frac{1}{2}k\right)}=(k+1)(k+3){\mu }_1,}$

${\displaystyle {\mu }_6=(k)(k+2)(k+4),}$

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

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

From these expressions we may derive the following relationships:

Mean: ${\displaystyle \mu =\sqrt{2}\frac{\mathrm{\Gamma }\left(\frac{1}{2}(k+1)\right)}{\mathrm{\Gamma }\left(\frac{1}{2}k\right)},}$ which is close to ${\displaystyle \sqrt{k-\frac{1}{2}}}$ for large k.

Variance: ${\displaystyle V=k-{\mu }^2,}$ which approaches ${\displaystyle \frac{1}{2}}$ as k increases.

Skewness: ${\displaystyle {\gamma }_1=\frac{\mu }{{\sigma }^3}\left(1-2{\sigma }^2\right).}$

Kurtosis excess: ${\displaystyle {\gamma }_2=\frac{2}{{\sigma }^2}\left(1-\mu \sigma {\gamma }_1-{\sigma }^2\right).}$

Entropy

The entropy is given by:

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

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

Large n approximation

We find the large n=k+1 approximation of the mean and variance of chi distribution. This has application e.g. in finding the distribution of standard deviation of a sample of normally distributed population, where n is the sample size.

The mean is then:

${\displaystyle \mu =\sqrt{2}\frac{\mathrm{\Gamma }(n/2)}{\mathrm{\Gamma }((n-1)/2)}}$

We use the Legendre duplication formula to write:

${\displaystyle 2^{n-2}\mathrm{\Gamma }((n-1)/2)\cdot \mathrm{\Gamma }(n/2)=\sqrt{\pi }\mathrm{\Gamma }(n-1)}$,

so that:

${\displaystyle \mu =\sqrt{2/\pi }{2}^{n-2}\frac{(\mathrm{\Gamma }(n/2){)}^2}{\mathrm{\Gamma }(n-1)}}$

Using Stirling's approximation for Gamma function, we get the following expression for the mean:

${\displaystyle \mu =\sqrt{2/\pi }{2}^{n-2}\frac{{\left(\sqrt{2\pi }(n/2-1{)}^{n/2-1+1/2}{e}^{-(n/2-1)}\cdot [1+\frac{1}{12(n/2-1)}+O(\frac{1}{n^2})]\right)}^2}{\sqrt{2\pi }(n-2{)}^{n-2+1/2}{e}^{-(n-2)}\cdot [1+\frac{1}{12(n-2)}+O(\frac{1}{n^2})]}}$

${\displaystyle =(n-2{)}^{1/2}\cdot \left[1+\frac{1}{4n}+O(\frac{1}{n^2})\right]=\sqrt{n-1}(1-\frac{1}{n-1}{)}^{1/2}\cdot \left[1+\frac{1}{4n}+O(\frac{1}{n^2})\right]}$

${\displaystyle =\sqrt{n-1}\cdot \left[1-\frac{1}{2n}+O(\frac{1}{n^2})\right]\cdot \left[1+\frac{1}{4n}+O(\frac{1}{n^2})\right]}$

${\displaystyle =\sqrt{n-1}\cdot \left[1-\frac{1}{4n}+O(\frac{1}{n^2})\right]}$

And thus the variance is:

${\displaystyle V=(n-1)-{\mu }^2=(n-1)\cdot \frac{1}{2n}\cdot \left[1+O(\frac{1}{n})\right]=\frac{1}{2}+O(\frac{1}{n})}$

Related distributions

• If ${\displaystyle X\sim {\chi }_k}$ then ${\displaystyle X^2\sim {\chi }_k^2}$ (chi-squared distribution)
• ${\displaystyle {\chi }_1\sim \mathrm{H}\mathrm{N}(1)}$ (half-normal distribution), i.e. if ${\displaystyle X\sim N(0,1)}$ then ${\displaystyle |X|\sim {\chi }_1}$, and if ${\displaystyle Y\sim \mathrm{H}\mathrm{N}(\sigma )}$ for any ${\displaystyle \sigma >0}$ then ${\displaystyle \frac{Y}{\sigma }\sim {\chi }_1}$
• ${\displaystyle {\chi }_2\sim \mathrm{R}\mathrm{a}\mathrm{y}\mathrm{l}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}(1)}$ (Rayleigh distribution) and if ${\displaystyle Y\sim \mathrm{R}\mathrm{a}\mathrm{y}\mathrm{l}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}(\sigma )}$ for any ${\displaystyle \sigma >0}$ then ${\displaystyle \frac{Y}{\sigma }\sim {\chi }_2}$
• ${\displaystyle {\chi }_3\sim \mathrm{M}\mathrm{a}\mathrm{x}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}(1)}$ (Maxwell distribution) and if ${\displaystyle Y\sim \mathrm{M}\mathrm{a}\mathrm{x}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}(a)}$ for any ${\displaystyle a>0}$ then ${\displaystyle \frac{Y}{a}\sim {\chi }_3}$
• ${\displaystyle \Vert {\mathit{N}}_{i=1,\dots ,k}(0,1){\Vert }_2\sim {\chi }_k}$, the Euclidean norm of a standard normal random vector of with ${\displaystyle k}$ dimensions, is distributed according to a chi distribution with ${\displaystyle k}$ degrees of freedom
• chi distribution is a special case of various distributions: generalized gamma, Nakagami, noncentral chi, etc.
• ${\displaystyle \underset{k\to \mathrm{\infty }}{lim}\frac{{\chi }_k-{\mu }_k}{{\sigma }_k}\stackrel{d}{\to }N(0,1)}$ (Normal distribution)
• The mean of the chi distribution (scaled by the square root of ${\displaystyle n-1}$) yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution.