Lévy distribution

Lévy (unshifted)
Lévy (unshifted)
	Probability density function;
	Cumulative distribution function;
Parameters	location; scale
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
Skewness	undefined
Excess kurtosis	undefined
Entropy	where is the Euler-Mascheroni constant
MGF	undefined
CF

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile. It is a special case of the inverse-gamma distribution. It is a stable distribution.

Lévy (unshifted)
Probability density function
Cumulative distribution function
Parameters	$\mu$ location; $c>0\,$ scale
Support	$x\in (\mu ,\infty )$
PDF	${\sqrt {\frac {c}{2\pi }}}~~{\frac {e^{-{\frac {c}{2(x-\mu )}}}}{(x-\mu )^{3/2}}}$
CDF	${\textrm {erfc}}\left({\sqrt {\frac {c}{2(x-\mu )}}}\right)$
Quantile	$\mu +{\frac {\sigma }{2\left({\textrm {erfc}}^{-1}(p)\right)^{2}}}$
Mean	$\infty$
Median	$\mu +c/2({\textrm {erfc}}^{-1}(1/2))^{2}\,$
Mode	$\mu +{\frac {c}{3}}$
Variance	$\infty$
Skewness	undefined
Excess kurtosis	undefined
Entropy	${\frac {1+3\gamma +\ln(16\pi c^{2})}{2}}$ where $\gamma$ is the Euler-Mascheroni constant
MGF	undefined
CF	$e^{i\mu t-{\sqrt {-2ict}}}$