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| {{Probability distribution|
| | Hello and welcome. My name is Irwin and I totally dig that name. To gather badges is what her family and her enjoy. Managing individuals is his profession. Puerto Rico is where he's been residing for many years and he will by no means transfer.<br><br>Also visit my blog post - [http://s-Wift.co/weightlossfoodprograms67803 meal delivery service] |
| name =chi|
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| type =density|
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| pdf_image =[[Image:Chi distribution PDF.svg|325px|Plot of the Rayleigh PMF]]<br /><small></small>|
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| cdf_image =[[Image:Chi distribution CDF.svg|325px|Plot of the Rayleigh CMF]]<br /><small></small>|
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| parameters =<math>k>0\,</math> (degrees of freedom)|
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| support =<math>x\in [0;\infty)</math>|
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| pdf =<math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>|
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| cdf =<math>P(k/2,x^2/2)\,</math>|
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| mean =<math>\mu=\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}</math>|
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| median =|
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| mode =<math>\sqrt{k-1}\,</math> for <math>k\ge 1</math>|
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| variance =<math>\sigma^2=k-\mu^2\,</math>|
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| skewness =<math>\gamma_1=\frac{\mu}{\sigma^3}\,(1-2\sigma^2)</math>|
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| kurtosis =<math>\frac{2}{\sigma^2}(1-\mu\sigma\gamma_1-\sigma^2)</math>|
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| entropy =<math>\ln(\Gamma(k/2))+\,</math><br /><math>\frac{1}{2}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_0(k/2))</math>|
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| mgf =Complicated (see text)|
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| char =Complicated (see text)|
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| }}
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| {{Unreferenced|date=October 2009}}
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| 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
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| :<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'' − 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>).
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| ==Characterization==
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| === Probability density function ===
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| The probability density function is
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| :<math>f(x;k) = \frac{2^{1-\frac{k}{2}}x^{k-1}e^{-\frac{x^2}{2}}}{\Gamma(\frac{k}{2})}</math>
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| where <math>\Gamma(z)</math> is the [[Gamma function]].
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| ===Cumulative distribution function===
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| The cumulative distribution function is given by:
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| :<math>F(x;k)=P(k/2,x^2/2)\,</math>
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| where <math>P(k,x)</math> is the [[regularized Gamma function]]. | |
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| ===Generating functions===
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| ==== Moment generating function ====
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| The [[moment generating function]] is given by:
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| :<math>M(t)=M\left(\frac{k}{2},\frac{1}{2},\frac{t^2}{2}\right)+</math>
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| :<math>t\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}
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| M\left(\frac{k+1}{2},\frac{3}{2},\frac{t^2}{2}\right)</math>
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| ====Characteristic function====
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| The [[Characteristic function (probability theory)|characteristic function]] is given by:
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| :<math>\varphi(t;k)=M\left(\frac{k}{2},\frac{1}{2},\frac{-t^2}{2}\right)+</math>
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| :<math>it\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}
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| M\left(\frac{k+1}{2},\frac{3}{2},\frac{-t^2}{2}\right)</math>
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| where again, <math>M(a,b,z)</math> is Kummer's [[confluent hypergeometric function]].
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| ==Properties==
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| === Moments ===
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| The raw [[moment (mathematics)|moments]] are then given by:
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| :<math>\mu_j=2^{j/2}\frac{\Gamma((k+j)/2)}{\Gamma(k/2)}</math>
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| where <math>\Gamma(z)</math> is the [[Gamma function]]. The first few raw moments are:
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| :<math>\mu_1=\sqrt{2}\,\,\frac{\Gamma((k\!+\!1)/2)}{\Gamma(k/2)}</math>
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| :<math>\mu_2=k\,</math>
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| :<math>\mu_3=2\sqrt{2}\,\,\frac{\Gamma((k\!+\!3)/2)}{\Gamma(k/2)}=(k+1)\mu_1</math>
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| :<math>\mu_4=(k)(k+2)\,</math>
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| :<math>\mu_5=4\sqrt{2}\,\,\frac{\Gamma((k\!+\!5)/2)}{\Gamma(k/2)}=(k+1)(k+3)\mu_1</math>
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| :<math>\mu_6=(k)(k+2)(k+4)\,</math>
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| where the rightmost expressions are derived using the recurrence relationship for the Gamma function:
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| :<math>\Gamma(x+1)=x\Gamma(x)\,</math>
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| From these expressions we may derive the following relationships:
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| Mean: <math>\mu=\sqrt{2}\,\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}</math>
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| Variance: <math>\sigma^2=k-\mu^2\,</math>
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| Skewness: <math>\gamma_1=\frac{\mu}{\sigma^3}\,(1-2\sigma^2)</math>
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| Kurtosis excess: <math>\gamma_2=\frac{2}{\sigma^2}(1-\mu\sigma\gamma_1-\sigma^2)</math>
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| ===Entropy===
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| The entropy is given by:
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| :<math>S=\ln(\Gamma(k/2))+\frac{1}{2}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_0(k/2))</math>
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| where <math>\psi_0(z)</math> is the [[polygamma function]].
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| ==Related distributions==
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| *If <math>X \sim \chi_k(x)</math> then <math>X^2 \sim \chi^2_k</math> ([[chi-squared distribution]])
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| *<math> \lim_{k \to \infty}\tfrac{\chi_k(x)-\mu_k}{\sigma_k} \xrightarrow{d}\ N(0,1) \,</math> ([[Normal distribution]])
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| *If <math> X \sim N(0,1)\,</math> then <math>| X | \sim \chi_1(x) \,</math>
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| *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>
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| *<math> \chi_2(x) \sim \mathrm{Rayleigh}(1)\,</math> ([[Rayleigh distribution]])
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| *<math> \chi_3(x) \sim \mathrm{Maxwell}(1)\,</math> ([[Maxwell distribution]])
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| *<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]])
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| *chi distribution is a special case of the [[generalized gamma distribution]] or the [[nakagami distribution]] or the [[noncentral chi distribution]]
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| <center>
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| {| class="wikitable"
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| |+ '''Various chi and chi-squared distributions'''
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| |-
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| ! Name !! Statistic
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| |-
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| | [[chi-squared distribution]] || <math>\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2</math>
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| |-
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| | [[noncentral chi-squared distribution]] || <math>\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2</math>
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| |-
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| | chi distribution || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>
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| |-
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| | [[noncentral chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}</math>
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| |}
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| </center>
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| ==See also==
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| *[[Nakagami distribution]]
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| ==External links==
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| * http://mathworld.wolfram.com/ChiDistribution.html
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{DEFAULTSORT:Chi Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Normal distribution]]
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| [[Category:Exponential family distributions]]
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| [[Category:Probability distributions]]
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