doc/source/tutorial/stats/continuous_skewcauchy.rst
.. _continuous-skew-cauchy:
This distribution is a generalization of the Cauchy distribution. It
has a single shape parameter :math:-1 < a < 1 that skews the distribution.
The special case :math:a=0 yields the Cauchy distribution.
.. math:: :nowrap:
\begin{eqnarray*}
f(x, a) & = & \frac{1}{\pi \left(\frac{x^2}{\left(a x + 1 \right)^2} + 1 \right)},\quad x\ge0; \\
& = & \frac{1}{\pi \left(\frac{x^2}{\left(-a x + 1 \right)^2} + 1 \right)},\quad x<0. \\
F(x, a) & = & \frac{1 - a}{2} + \frac{1 + a}{\pi} \arctan\left(\frac{x}{1 + a} \right),\quad x\ge0; \\
& = & \frac{1 - a}{2} + \frac{1 - a}{\pi} \arctan\left(\frac{x}{1 - a} \right),\quad x<0.
\end{eqnarray*}
The mean, variance, skewness, and kurtosis are all undefined.
Implementation: scipy.stats.skewcauchy