HalfCauchy Distribution

If \(Z\) is Hyperbolic Secant distributed then \(e^{Z}\) is Half-Cauchy distributed. Also, if \(W\) is (standard) Cauchy distributed, then \(\left|W\right|\) is Half-Cauchy distributed. Special case of the Folded Cauchy distribution with \(c=0.\) The support is \(x\geq0\). The standard form is

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

\[M\left(t\right)=\cos t+\frac{2}{\pi}\left[\mathrm{Si}\left(t\right)\cos t-\mathrm{Ci}\left(\mathrm{-}t\right)\sin t\right]\]

where \(\mathrm{Si}(t)=\int_0^t \frac{\sin x}{x} dx\), \(\mathrm{Ci}(t)=-\int_t^\infty \frac{\cos x}{x} dx\).

\begin{eqnarray*} m_{d} & = & 0\\ m_{n} & = & \tan\left(\frac{\pi}{4}\right)\end{eqnarray*}

No moments, as the integrals diverge.

\begin{eqnarray*} h\left[X\right] & = & \log\left(2\pi\right)\\ & \approx & 1.8378770664093454836.\end{eqnarray*}

Implementation: scipy.stats.halfcauchy