Chi Distribution

The probability density function and cumulative distribution function are

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

$$D_n(x) = Q({\textstyle{1\over 2}}n, {\textstyle{1\over 2}}x^2),$$ (2)

where $Q$ is the Regularized Gamma Function.

$$\mu = {\sqrt{2}\,\Gamma({\textstyle{1\over 2}}(n+1))\over \Gamma({\textstyle{1\over 2}}n)}$$ (3)

$$\sigma^2 = {2[\Gamma({\textstyle{1\over 2}}n)\Gamma(1+{\textstyle{1\over 2}}n)-\Gamma^2({\textstyle{1\over 2}}(n+1))]\over\Gamma^2({\textstyle{1\over 2}}n)}$$ (4)

$$\gamma_1 = {2\Gamma^3({\textstyle{1\over 2}}(n+1))-3\Gamma({\textstyle{1\ove... ...\Gamma(1+{\textstyle{1\over 2}}n)-\Gamma^2({\textstyle{1\over 2}}(n+1))]^{3/2}}$$

$$\phantom{=} +{\Gamma^2({\textstyle{1\over 2}}n)\Gamma({\textstyle{3+n\over 2}... ...\Gamma(1+{\textstyle{1\over 2}}n)-\Gamma^2({\textstyle{1\over 2}}(n+1))]^{3/2}}$$ (5)

$$\gamma_2 = {-3\Gamma^4({\textstyle{1\over 2}}(n+1))+6\Gamma({\textstyle{1\ov... ...2}}n)\Gamma({\textstyle{2+n\over 2}})-\Gamma^2({\textstyle{1\over 2}}(n+1))]^2}$$

$$\phantom{=} +{-4\Gamma^2({\textstyle{1\over 2}}n)\Gamma({\textstyle{1\over 2}... ...}}n)\Gamma({\textstyle{2+n\over 2}})-\Gamma^2({\textstyle{1\over 2}}(n+1))]^2},$$ (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