# Skewed Cauchy Distribution

This distribution is a generalization of the Cauchy distribution. It has a single shape parameter \(-1 < a < 1\) that skews the distribution. The special case \(a=0\) yields the Cauchy distribution.

## Functions

\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.

## References

- “Skewed generalized t distribution”, Wikipedia https://en.wikipedia.org/wiki/Skewed_generalized_t_distribution#Skewed_Cauchy_distribution

Implementation: `scipy.stats.skewcauchy`