Lévy Distribution

A special case of Lévy-stable distributions with \(\alpha=\frac{1}{2}\) and \(\beta=1\) . In standard form it is defined for \(x>0\) as

\[ \begin{eqnarray*} f\left(x\right) & = & \frac{1}{x\sqrt{2\pi x}}\exp\left(-\frac{1}{2x}\right)\\ F\left(x\right) & = & 2\left[1-\Phi\left(\frac{1}{\sqrt{x}}\right)\right]\\ G\left(q\right) & = & \left[\Phi^{-1}\left(1-\frac{q}{2}\right)\right]^{-2}.\end{eqnarray*}\]

It has no finite moments.

Implementation: scipy.stats.levy