Skew normal distribution

Skew Normal
Skew Normal
	Probability density function
	Cumulative distribution function
Parameters	location (real); scale (positive, real); shape (real)
Support
PDF
CDF	; is Owen's T function
Mean	where
Mode
Variance
Skewness
Excess kurtosis
MGF
CF

In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

Skew Normal
Probability density function
Cumulative distribution function
Parameters	$\xi \,$ location (real) $\omega \,$ scale (positive, real) $\alpha \,$ shape (real)
Support	${\displaystyle x\in (-\infty$
PDF	${\frac {2}{\omega {\sqrt {2\pi }}}}e^{-{\frac {(x-\xi )^{2}}{2\omega ^{2}}}}\int _{-\infty }^{\alpha \left({\frac {x-\xi }{\omega }}\right)}{\frac {1}{\sqrt {2\pi }}}e^{-{\frac {t^{2}}{2}}}\ \mathrm {d} t$
CDF	$\Phi \left({\frac {x-\xi }{\omega }}\right)-2T\left({\frac {x-\xi }{\omega }},\alpha \right)$ $T(h,a)$ is Owen's T function
Mean	$\xi +\omega \delta {\sqrt {\frac {2}{\pi }}}$ where $\delta ={\frac {\alpha }{\sqrt {1+\alpha ^{2}}}}$
Mode	$\xi +\omega m_{o}(\alpha )$
Variance	$\omega ^{2}\left(1-{\frac {2\delta ^{2}}{\pi }}\right)$
Skewness	$\gamma _{1}={\frac {4-\pi }{2}}{\frac {\left(\delta {\sqrt {2/\pi }}\right)^{3}}{\left(1-2\delta ^{2}/\pi \right)^{3/2}}}$
Excess kurtosis	$2(\pi -3){\frac {\left(\delta {\sqrt {2/\pi }}\right)^{4}}{\left(1-2\delta ^{2}/\pi \right)^{2}}}$
MGF	$M_{X}\left(t\right)=2\exp \left(\xi t+{\frac {\omega ^{2}t^{2}}{2}}\right)\Phi \left(\omega \delta t\right)$
CF	$e^{it\xi -t^{2}\omega ^{2}/2}\left(1+i\,{\textrm {Erfi}}\left({\frac {\delta \omega t}{\sqrt {2}}}\right)\right)$