# Stochastic Extreme Wind Speed Modeling and Bayes Estimation under the Inverse Rayleigh Distribution
## Appendix C The Gamma and Negative Log-Gamma Distributions  
The gamma PDF with positive parameters \(v\) and \(\delta\) shape and scale parameters respectively is expressed by the following:  
\[f(x)=\frac{1}{\delta^{v}\Gamma(v)}x^{v-1}e^{-(\frac{x}{\delta})} \tag{11}\]  
where \(\Gamma(\alpha)\) is the \"Euler-gamma\" function evaluated at \(\alpha\). The mean value and variance of the distribution are as follows:  
\[\mathrm{E}[X]=v\,\delta;\;\mathrm{Var}[X]=v\,\delta^{2} \tag{12}\]  
Then, if \(\mathrm{Y}=\mathrm{e}^{-X}\), the RV \(\mathrm{Y}\) has a negative log-gamma (NLGD) distribution, for which the PDF is:  
\[f(y)=\frac{1}{\delta^{v}\Gamma(v)}y^{[(\frac{1}{\delta})-1]}(-\log y)^{v-1}, \;0\leq y\leq 1 \tag{13}\]  
with \(v\) and \(\delta\) being the above gamma PDF parameters. The NLGD mean value and variance, are, respectively, equal to the following:  
\[\mu_{NLG}=\frac{1}{\left(1+\delta\right)^{v}};\;\sigma^{2}_{NLG}=\frac{1}{ \left(1+2\delta\right)^{v}}-\frac{1}{\left(1+\delta\right)^{2v}} \tag{14}\]The \(k\)th moment of the NLGD PDF is as follows:  
\[\mu_{k}=\mathbb{E}\Big{[}\mathbb{Y}^{k}\Big{]}=\frac{1}{\left(1+k\delta\right)^{ v}} \tag{10}\]  
Appendix D The Beta distribution and its transformations: Complementary Beta and Negative Exponential Beta Distributions  
A RV Y--assuming values on (0,1)--has a _beta_ distribution if its PDF is expressible as follows:  
\[f(y;p,q)=betapdf(y;p,q)=\left\{ \begin{array}{cc}\Gamma(p+q)\frac{\Psi^{-1}(1 -y)^{t-1}}{\Gamma(p)\Gamma(q)},&0<y<1\\ 0&elsewhere\end{array} \right. \tag{11}\]  
where \(p\) and \(q\) are positive shape parameters. In particular, if \(p=q=1\), a uniform distribution over (0,1) is obtained. The mean value and the variance of the beta distribution are given by the following:  
\[\mu=\frac{p}{p+q};\;\sigma^{2}=\mu^{2}\frac{q}{p(p+q+1)} \tag{12}\]  
The above model is denoted as the \"a beta(\(p\),\(q\))\" distribution. Some properties and transformations are reported in the following. It is easy to show that, if Y is a beta(\(p\),\(q\)) RV, then the following \"complementary beta\" RV X = 1-Y is a beta(\(q\),\(p\)) RV. Let Y be a beta(\(a\),\(b\)) RV. The PDF of the transformed RV X = -log(Y), here denoted as a Negative Exponential Beta Distribution (NEBD) PDF, by well-known rules of RV transformations [31] is as follows:  
\[g_{k}(x)=negebpdf(x;a,b)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}e^{-ax}(1-e^{- x})^{(b-1)},\;0<x<\infty \tag{13}\]  
Another way to look at this model, very useful for the problem under study, is that the RV Y = e\({}^{-X}\) has a beta(a,b) distribution if and only its inverse function X = -log(Y)\({}^{\prime\prime}\) has a NEBD(\(a\),\(b\)) PDF. Mean and variance of this model appear not easy to evaluate, but since X = -log(R) is a decreasing function of Y, it is possible to immediately compute the \(p\)-quantile of X, X\({}_{\rm p}\), by using the q-quantile of Y, where \(q=1-p\):  
\[\mathbb{X}_{\rm p}=-\text{logY}_{\text{(1-p)}}\]  
Appendix E Relationship between the Beta and the Gamma distribution and the distribution of the product of the Gamma and Beta RV  
Let X and Y be two independent RVs, X characterized by a gamma, and Y characterized by a beta distribution. \[f(x)=\frac{1}{8^{v}\Gamma(v)}x^{v-1}e^{-(\frac{x}{v})} \tag{14}\]  
\[f(y;p,q)=betapdf(y;p,q)=\left\{ \begin{array}{cc}\Gamma(p+q)\frac{\Psi^{-1}( 1-y)^{t+1}}{\Gamma(p)\Gamma(q)},&0<y<1\\ 0&elsewhere\end{array} \right. \tag{15}\]  
Let us consider the product RV: \(Z=XY\). Because of the independence, its mean value is equal to  
\[\mathbb{E}[Z]=\mathbb{E}[X]\mathbb{E}[Y]=v\delta\frac{p}{p+q} \tag{16}\]  
Its PDF [37] can be evaluated as (using the transformation \(\left[ \begin{array}{c}Z\\ U\end{array} \right]=\left[ \begin{array}{c}XY\\ X\end{array} \right]\))  
\[f(z)=\int\limits_{z}^{\infty}u^{v-1}e^{-u\delta}\Big{(}\frac{z}{u}\Big{)}^{p- 1}\Big{(}1-\frac{z}{u}\Big{)}^{q-1}z^{-1}du \tag{17}\]  
This integral may be expressed in terms of confluent hypergeometric functions [37].