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.. _continuous-cauchy:
Cauchy Distribution
===================
The support is :math:`x\in\mathbb{R}`.
.. math::
:nowrap:
\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 :math:`t` distribution with one degree of
freedom.
.. math::
:nowrap:
\begin{eqnarray*} h\left[X\right] & = & \log\left(4\pi\right)\\ & \approx & 2.5310242469692907930.\end{eqnarray*}
Implementation: `scipy.stats.cauchy`
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