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{{Probability distribution|
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  name      =chi|
  type      =density|
  pdf_image =[[Image:Chi distribution PDF.svg|325px|Plot of the Rayleigh PMF]]<br /><small></small>|
  cdf_image  =[[Image:Chi distribution CDF.svg|325px|Plot of the Rayleigh CMF]]<br /><small></small>|
  parameters =<math>k>0\,</math> (degrees of freedom)|
  support    =<math>x\in [0;\infty)</math>|
  pdf        =<math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>|
  cdf        =<math>P(k/2,x^2/2)\,</math>|
  mean      =<math>\mu=\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}</math>|
  median    =|
  mode      =<math>\sqrt{k-1}\,</math> for <math>k\ge 1</math>|
  variance  =<math>\sigma^2=k-\mu^2\,</math>|
  skewness  =<math>\gamma_1=\frac{\mu}{\sigma^3}\,(1-2\sigma^2)</math>|
  kurtosis  =<math>\frac{2}{\sigma^2}(1-\mu\sigma\gamma_1-\sigma^2)</math>|
  entropy    =<math>\ln(\Gamma(k/2))+\,</math><br /><math>\frac{1}{2}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_0(k/2))</math>|
  mgf        =Complicated (see text)|
  char      =Complicated (see text)|
}}
{{Unreferenced|date=October 2009}}
In [[probability theory]] and [[statistics]], the '''chi distribution''' is a continuous [[probability distribution]].  It is the distribution of the square root of the sum of squares of independent random variables having a standard [[normal distribution]]. The most familiar example is the [[Maxwell distribution]] of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom (one for each spatial coordinate). If <math>X_i</math> are ''k'' independent, [[normal distribution|normally distributed]] random variables with means <math>\mu_i</math> and [[standard deviation]]s <math>\sigma_i</math>, then the statistic


:<math>Y = \sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>
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is distributed according to the chi distribution. Accordingly, dividing by the mean of the chi distribution (scaled by the square root of ''n''&nbsp;&minus;&nbsp;1) yields the correction factor in the [[Unbiased estimation of standard deviation#Results for the normal distribution|unbiased estimation of the standard deviation of the normal distribution]]. The chi distribution has one parameter: <math>k</math> which specifies the number of [[Degrees of freedom (statistics)|degrees of freedom]] (i.e. the number of <math>X_i</math>).
 
==Characterization==
 
=== Probability density function ===
The probability density function is
:<math>f(x;k) = \frac{2^{1-\frac{k}{2}}x^{k-1}e^{-\frac{x^2}{2}}}{\Gamma(\frac{k}{2})}</math>
 
where <math>\Gamma(z)</math> is the [[Gamma function]].
 
===Cumulative distribution function===
The cumulative distribution function is given by:
 
:<math>F(x;k)=P(k/2,x^2/2)\,</math>
 
where <math>P(k,x)</math> is  the [[regularized Gamma function]].
 
===Generating functions===
 
==== Moment generating function ====
The [[moment generating function]] is given by:
 
:<math>M(t)=M\left(\frac{k}{2},\frac{1}{2},\frac{t^2}{2}\right)+</math>
:<math>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)</math>
 
====Characteristic function====
The [[Characteristic function (probability theory)|characteristic function]] is given by:
 
:<math>\varphi(t;k)=M\left(\frac{k}{2},\frac{1}{2},\frac{-t^2}{2}\right)+</math>
:<math>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)</math>
 
where again, <math>M(a,b,z)</math> is Kummer's [[confluent hypergeometric function]].
 
==Properties==
 
=== Moments ===
The raw [[moment (mathematics)|moments]] are then given by:
 
:<math>\mu_j=2^{j/2}\frac{\Gamma((k+j)/2)}{\Gamma(k/2)}</math>
 
where <math>\Gamma(z)</math> is the [[Gamma function]]. The first few raw moments are:
 
:<math>\mu_1=\sqrt{2}\,\,\frac{\Gamma((k\!+\!1)/2)}{\Gamma(k/2)}</math>
:<math>\mu_2=k\,</math>
:<math>\mu_3=2\sqrt{2}\,\,\frac{\Gamma((k\!+\!3)/2)}{\Gamma(k/2)}=(k+1)\mu_1</math>
:<math>\mu_4=(k)(k+2)\,</math>
:<math>\mu_5=4\sqrt{2}\,\,\frac{\Gamma((k\!+\!5)/2)}{\Gamma(k/2)}=(k+1)(k+3)\mu_1</math>
:<math>\mu_6=(k)(k+2)(k+4)\,</math>
 
where the rightmost expressions are derived using the recurrence relationship for the Gamma function:
 
:<math>\Gamma(x+1)=x\Gamma(x)\,</math>
 
From these expressions we may derive the following relationships:
 
Mean: <math>\mu=\sqrt{2}\,\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}</math>
 
Variance: <math>\sigma^2=k-\mu^2\,</math>
 
Skewness: <math>\gamma_1=\frac{\mu}{\sigma^3}\,(1-2\sigma^2)</math>
 
Kurtosis excess: <math>\gamma_2=\frac{2}{\sigma^2}(1-\mu\sigma\gamma_1-\sigma^2)</math>
 
===Entropy===
The entropy is given by:
 
:<math>S=\ln(\Gamma(k/2))+\frac{1}{2}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_0(k/2))</math>
 
where <math>\psi_0(z)</math> is the [[polygamma function]].
 
==Related distributions==
*If <math>X \sim \chi_k(x)</math> then <math>X^2 \sim \chi^2_k</math> ([[chi-squared distribution]])
*<math> \lim_{k \to \infty}\tfrac{\chi_k(x)-\mu_k}{\sigma_k}  \xrightarrow{d}\ N(0,1) \,</math> ([[Normal distribution]])
*If <math> X \sim N(0,1)\,</math> then <math>| X | \sim \chi_1(x) \,</math>
*If <math>X \sim \chi_1(x) \,</math> then <math>\sigma X \sim HN(\sigma)\,</math> ([[half-normal distribution]]) for any <math> \sigma > 0 \, </math>
*<math> \chi_2(x) \sim \mathrm{Rayleigh}(1)\,</math> ([[Rayleigh distribution]])
*<math> \chi_3(x) \sim \mathrm{Maxwell}(1)\,</math> ([[Maxwell distribution]])
*<math> \|\boldsymbol{N}_{i=1,\ldots,k}{(0,1)}\|_2 \sim \chi_k(x) </math> (The [[Norm (mathematics)#Euclidean norm|2-norm]] of <math> k </math> standard normally distributed variables is a chi distribution with <math> k </math> [[Degrees of freedom (statistics)|degrees of freedom]])
*chi distribution is a special case of the [[generalized gamma distribution]] or the [[nakagami distribution]] or the [[noncentral chi distribution]]
 
<center>
{| class="wikitable"
|+ '''Various chi and chi-squared distributions'''
|-
! Name !! Statistic
|-
| [[chi-squared distribution]] || <math>\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2</math>
|-
| [[noncentral chi-squared distribution]] || <math>\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2</math>
|-
| chi distribution || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>
|-
| [[noncentral chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}</math>
|}
</center>
 
==See also==
*[[Nakagami distribution]]
 
==External links==
* http://mathworld.wolfram.com/ChiDistribution.html
{{ProbDistributions|continuous-semi-infinite}}
 
{{DEFAULTSORT:Chi Distribution}}
[[Category:Continuous distributions]]
[[Category:Normal distribution]]
[[Category:Exponential family distributions]]
[[Category:Probability distributions]]

Revision as of 18:44, 28 February 2014

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