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Chi Distribution

The probability density function and cumulative distribution function are

$\displaystyle P_n(x)$ $\textstyle =$ $\displaystyle {2^{1-n/2}x^{n-1}e^{-x^2/2}\over\Gamma({\textstyle{1\over 2}}n)}$ (1)
$\displaystyle D_n(x)$ $\textstyle =$ $\displaystyle Q({\textstyle{1\over 2}}n, {\textstyle{1\over 2}}x^2),$ (2)

where $Q$ is the Regularized Gamma Function.


$\displaystyle \mu$ $\textstyle =$ $\displaystyle {\sqrt{2}\,\Gamma({\textstyle{1\over 2}}(n+1))\over \Gamma({\textstyle{1\over 2}}n)}$ (3)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle {2[\Gamma({\textstyle{1\over 2}}n)\Gamma(1+{\textstyle{1\over 2}}n)-\Gamma^2({\textstyle{1\over 2}}(n+1))]\over\Gamma^2({\textstyle{1\over 2}}n)}$ (4)
$\displaystyle \gamma_1$ $\textstyle =$ $\displaystyle {2\Gamma^3({\textstyle{1\over 2}}(n+1))-3\Gamma({\textstyle{1\ove...
...\Gamma(1+{\textstyle{1\over 2}}n)-\Gamma^2({\textstyle{1\over 2}}(n+1))]^{3/2}}$  
  $\textstyle \phantom{=}$ $\displaystyle +{\Gamma^2({\textstyle{1\over 2}}n)\Gamma({\textstyle{3+n\over 2}...
...\Gamma(1+{\textstyle{1\over 2}}n)-\Gamma^2({\textstyle{1\over 2}}(n+1))]^{3/2}}$ (5)
$\displaystyle \gamma_2$ $\textstyle =$ $\displaystyle {-3\Gamma^4({\textstyle{1\over 2}}(n+1))+6\Gamma({\textstyle{1\ov...
...2}}n)\Gamma({\textstyle{2+n\over 2}})-\Gamma^2({\textstyle{1\over 2}}(n+1))]^2}$  
  $\textstyle \phantom{=}$ $\displaystyle +{-4\Gamma^2({\textstyle{1\over 2}}n)\Gamma({\textstyle{1\over 2}...
...}}n)\Gamma({\textstyle{2+n\over 2}})-\Gamma^2({\textstyle{1\over 2}}(n+1))]^2},$ (6)

where $\mu$ is the Mean, $\sigma^2$ the Variance, $\gamma_1$ the Skewness, and $\gamma_2$ the Kurtosis. For $n=1$, the $\chi$ distribution is a Half-Normal Distribution with $\theta=1$. For $n=2$, it is a Rayleigh Distribution with $\sigma=1$.

See also Chi-Squared Distribution, Half-Normal Distribution, Rayleigh Distribution




© 1996-9 Eric W. Weisstein
1999-05-26