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; \right) & = &\frac{\beta}{2\Gamma(1/\beta)} e^{-\left|x\right|^{\beta}} \end{eqnarray*}
\begin{eqnarray*} F\left(x; \right) & = & \frac{1}{2} + \rm{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; \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 \([-1, 1]\) (\(\beta \rightarrow \infty\))