doc/source/tutorial/stats/continuous_gennorm.rst
.. _continuous-gennorm:
This distribution is also known as the exponential power distribution. It has
a single shape parameter :math:\beta>0. It reduces to a number of common
distributions. The support is :math:x \in (-\infty, \infty).
.. math:: :nowrap:
\begin{eqnarray*} f\left(x; \beta\right) & = &\frac{\beta}{2\Gamma(1/\beta)} e^{-\left|x\right|^{\beta}} \end{eqnarray*}
.. math:: :nowrap:
\begin{eqnarray*} F\left(x; \beta\right) & = & \frac{1}{2} + \operatorname{sgn}\left(x\right) \frac{\gamma\left(1/\beta, \left|x\right|^{\beta}\right)}{2\Gamma\left(1/\beta\right)} \end{eqnarray*}
where :math:\gamma\left(s, x\right) = \int_0^x t^{s-1} e^{-t} dt is the lower incomplete gamma function.
.. math:: :nowrap:
\begin{eqnarray*} h\left[X; \beta\right] = \frac{1}{\beta} - \log\left(\frac{\beta}{2\Gamma\left(1/\beta\right)}\right)\end{eqnarray*}
.. math:: :nowrap:
\begin{eqnarray*}
\mu & = & 0 \\
m_{n} & = & 0 \\
m_{d} & = & 0 \\
\mu_2 & = & \frac{\Gamma\left(3/\beta\right)}{\Gamma\left(1/\beta\right)} \\
\gamma_1 & = & 0 \\
\gamma_2 & = & \frac{\Gamma\left(5/\beta\right) \Gamma\left(1/\beta\right)}{\Gamma\left(3/\beta\right)^2} - 3 \\
\end{eqnarray*}
\beta = 1)\frac{1}{\sqrt{2}} (:math:\beta = 2)[-1, 1] (:math:\beta \rightarrow \infty)Implementation: scipy.stats.gennorm