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Cauchy Distribution

The support is \(x\in\mathbb{R}\).

\begin{eqnarray*} f\left(x\right) & = & \frac{1}{\pi\left(1+x^{2}\right)}\\ F\left(x\right) & = & \frac{1}{2}+\frac{1}{\pi}\tan^{-1}x\\ G\left(q\right) & = & \tan\left(\pi q-\frac{\pi}{2}\right)\\ m_{d} & = & 0\\ m_{n} & = & 0\end{eqnarray*}

No finite moments. This is the \(t\) distribution with one degree of freedom.

\begin{eqnarray*} h\left[X\right] & = & \log\left(4\pi\right)\\ & \approx & 2.5310242469692907930.\end{eqnarray*}

Implementation: scipy.stats.cauchy