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{array}{rcl} f (x) & = & \frac{2}{π (1 + x^{2})} \\ F (x) & = & \frac{2}{π} \arctan (x) \\ G (q) & = & \tan (\frac{π}{2} q) \end{array}

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

M (t) = \cos t + \frac{2}{π} [S i (t) \cos t - C i (- t) \sin t]

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

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

\begin{array}{rcl} m_{d} & = & 0 \\ m_{n} & = & \tan (\frac{π}{4}) \end{array}

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

No moments, as the integrals diverge.

\begin{array}{rcl} h [X] & = & \log (2 π) \\ \approx & 1.8378770664093454836. \end{array}

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

Implementation: scipy.stats.halfcauchy