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chi

Probability density function Image:Chi distribution PDF.png
Cumulative distribution function Image:Chi distribution CDF.png
Parameters	<math>k>0\,</math> (degrees of freedom)
Support	<math>x\in [0;\infty)</math>
Probability density function (pdf)	<math>\frac{2^{1-k/2}x^{k-1}e^{-x^2/2
Cumulative distribution function (cdf)	{{{cdf}}}
Mean	{{{mean}}}
Median	{{{median}}}
Mode	{{{mode}}}
Variance	{{{variance}}}
Skewness	{{{skewness}}}
Excess Kurtosis	{{{kurtosis}}}
Entropy	{{{entropy}}}
mgf	{{{mgf}}}
Char. func.	{{{char}}}

{\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>
<math>\frac{1}{2}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_0(k/2))</math>|
 mgf        =Complicated (see text)|
 char       =Complicated (see text)|

}}

In probability theory and statistics, the chi distribution is a continuous probability distribution. The distribution usually arises when a k-dimensional vector's orthogonal components are independent and each follow a standard normal distribution. The length of the vector will then have a chi distribution. The most familiar example is the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom. If <math>X_i</math> are k independent, normally distributed random variables with means <math>\mu_i</math> and standard deviations <math>\sigma_i</math>, then the statistic

<math>Z = \sqrt{\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>

is distributed according to the chi distribution. The chi distribution has one parameter: <math>k</math> which specifies the number of degrees of freedom (i.e. the number of <math>X_i</math>).

[edit] Properties

The probability density function is

<math>f(x;k) = \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>

where <math>\Gamma(z)</math> is the Gamma 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. 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>

where <math>M(a,b,z)</math> is Kummer's confluent hypergeometric function. The raw 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>

The 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. 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.

[edit] Related distributions

• If <math>X</math> is chi distributed <math>X \sim \chi_k(x)</math> then <math>X^2</math> is chi-square distributed: <math>X^2 \sim \chi^2_k</math>
• The Rayleigh distribution with <math>\sigma=1</math> is a chi distribution with two degrees of freedom.
• The Maxwell distribution for normalized molecular speeds is a chi distribution with three degrees of freedom.
• The chi distribution for <math>k=1</math> is the half-normal distribution.

Various chi and chi-square distributions

Name	Statistic
chi-square distribution	<math>\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2</math>
noncentral chi-square distribution	<math>\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2</math>
chi distribution	<math>\sqrt{\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>
noncentral chi distribution	<math>\sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}</math>