2.1.1.18. Generalized normal law

This law describes a generalized normal distribution depending on the location \(\mu\), the scale \(\alpha\) and the shape \(\beta\), as

\[f(x) = \frac{\beta}{2 \alpha \Gamma \left(1/\beta \right) } \times e^{-\left(\frac{x-\mu}{\alpha} \right)^{\beta}}\]

The mean value of the generalized normal law is \(\mu\) while its variance can be written as \(\sigma^2 = \frac{\alpha^{2} \Gamma(3/\beta)}{\Gamma(1/\beta) }\)

Figure 2.19 shows the PDF, CDF and inverse CDF generated for different sets of parameters.

Figure 2.19 Example of PDF, CDF and inverse CDF for GeneralizedNormal distributions.