# 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

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Left-skewed Lévy Distribution

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Logistic (Sech-squared) Distribution