This is documentation for an old release of SciPy (version 1.5.4). Read this page in the documentation of the latest stable release (version 1.15.1).
Generalized Normal Distribution¶
This distribution is also known as the exponential power distribution. It has a single shape parameter \(\beta>0\). It reduces to a number of common distributions.
Functions¶
\begin{eqnarray*} f\left(x; \beta\right) & = &\frac{\beta}{2\Gamma(1/\beta)} e^{-\left|x\right|^{\beta}} \end{eqnarray*}
\begin{eqnarray*} F\left(x; \beta\right) & = & \frac{1}{2} + \mathrm{sgn}\left(x\right) \frac{\gamma\left(1/\beta, x^{\beta}\right)}{2\Gamma\left(1/\beta\right)} \end{eqnarray*}
\(\gamma\) is the lower incomplete gamma function. \(\gamma\left(s, x\right) = \int_0^x t^{s-1} e^{-t} dt\).
\begin{eqnarray*} h\left[X; \beta\right] = \frac{1}{\beta} - \log\left(\frac{\beta}{2\Gamma\left(1/\beta\right)}\right)\end{eqnarray*}
Moments¶
\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*}
Special Cases¶
Laplace distribution (\(\beta = 1\))
Normal distribution with \(\mu_2 = 1/2\) (\(\beta = 2\))
Uniform distribution over the interval \([-1, 1]\) (\(\beta \rightarrow \infty\))