# Hyperbolic distribution

Parameters $\mu$ location (real) $\alpha$ (real) $\beta$ asymmetry parameter (real) $\delta$ scale parameter (real) $\gamma = \sqrt{\alpha^2 - \beta^2}$ $x \in (-\infty; +\infty)\!$ $\frac{\gamma}{2\alpha\delta K_1(\delta \gamma)} \; e^{-\alpha\sqrt{\delta^2 + (x - \mu)^2}+ \beta (x - \mu)}$ $K_\lambda$ denotes a modified Bessel function of the second kind $\mu + \frac{\delta \beta K_{2}(\delta \gamma)}{\gamma K_1(\delta\gamma)}$ $\mu + \frac{\delta\beta}{\gamma}$ $\frac{\delta K_{2}(\delta \gamma)}{\gamma K_1(\delta\gamma)} + \frac{\beta^2\delta^2}{\gamma^2}\left(\frac{K_{3}(\delta\gamma)}{K_{1}(\delta\gamma)} -\frac{K_{2}^2(\delta\gamma)}{K_{1}^2(\delta\gamma)} \right)$ $\frac{e^{\mu z}\gamma K_1(\delta \sqrt{ (\alpha^2 -(\beta +z)^2)})}{\sqrt{(\alpha^2 -(\beta +z)^2)}K_1 (\delta \gamma)}$