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# 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. The support is \(x \in (-\infty, \infty)\).

## 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} + \operatorname{sgn}\left(x\right) \frac{\gamma\left(1/\beta, \left|x\right|^{\beta}\right)}{2\Gamma\left(1/\beta\right)} \end{eqnarray*}

where \(\gamma\left(s, x\right) = \int_0^x t^{s-1} e^{-t} dt\) is the lower incomplete gamma function.

\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 with scale \(\frac{1}{\sqrt{2}}\) (\(\beta = 2\))

- Uniform distribution over the interval \([-1, 1]\) (\(\beta \rightarrow \infty\))

## Sources

- https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1

- https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function

Implementation: `scipy.stats.gennorm`