# Stochastic Extreme Wind Speed Modeling and Bayes Estimation under the Inverse Rayleigh Distribution
## 4 A Bayes Method of Inference for the Risk Index Based upon the IRD Model
### The Case of a Gamma Prior PDF on the IRD Parameter \(\alpha\): Prior Analysis  
First, for sake of numerical illustration in Table 1, some values of the RI, computed as  
\[\left\{ \begin{array}{c c c}R=R(t,z,\phi,\alpha)=1-e^{-\mathrm{WS}}\\ \mathrm{W}=\mathrm{pt}\\ S=1-F(z|\alpha)=1-e^{-(\frac{\alpha}{z^{2}})}\end{array} \right. \tag{45}\]  
are shown with reference to a fixed time interval of 1 year, a threshold value \(z=22\) m/s., and nine couples of values (\(\phi\), \(\alpha\)), shown respectively in the headings of the rows and columns of Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{\(\phi\)**(year–1)**} & \multicolumn{3}{c}{\(\alpha\)**(m2/s2)**} \\ \cline{2-4}  & **15** & **45** & **90** \\ \hline
2.5 & **0.0735** & 0.1991 & 0.3457 \\
5 & 0.1415 & **0.3585** & 0.5719 \\
10 & 0.2630 & 0.5885 & **0.8167** \\ \hline \end{tabular}
\end{table} 
Table 1: Some values of the RI corresponding to nine couples of values (\(\phi\), \(\alpha\)), \(t=1\) year, and \(z=22\). Also, the following numerical applications will refer to these typical values of the RI and parameters \(\phi\) and \(\alpha\). As already discussed, the values of the RI increase as both \(\phi\) and \(\alpha\) increase, so they increase from left to right, looking through rows, and from top to bottom, looking through columns. In particular, for the sake of brevity, the three cases corresponding to boldface characters in the principal diagonal of Table 1 will be those taken as references in the course of the following numerical applications. Considering that the parameters \(\phi\) and \(\alpha\) are RVs in the Bayes method, their values are to be considered expected values of their prior distributions. Therefore, the following three cases are taken as references:  
**Case A**: (\"low risk\" values): E[\(\phi\)] = 2.5, E[\(\alpha\)] = 15, corresponding to an approximate expected value of the RI: _E[R] \(\approx\) 0.0735_;
**Case B**: (\"medium risk\" values): E[\(\phi\)] = 5, E[\(\alpha\)] = 45, corresponding to an approximate expected value of the RI: _E[R] \(\approx\) 0.3585_;
**Case C**: (\"high risk\" values): E[\(\phi\)] = 10, E[\(\alpha\)] = 90, corresponding to an approximately expected value of the RI: _E[R] \(\approx\) 0.8167_  
The above reported expected values of the RI: (E[R] = _0.0735_, _0.3585_, and _0.8167_), as taken from Table 1, are merely indicative; indeed they are approximate expected values since the RI is a nonlinear function of (\(\phi\), \(\alpha\)):  
\[\text{E}\left[R\left(\phi,\,\alpha\right)\right]\
eq\text{R}\left[\left(\text {E}\left[\phi\right],\,\text{E}\left[\alpha\right]\right)\right]\]  
Indeed, for a given prior PDF \(p(\phi\), \(\alpha)\) of (\(\phi\), \(\alpha\)) which--as long as t = 1 year and \(W=\phi\) --coincides with the prior PDF \(p(w,\alpha)\) of (\(w,\alpha\)), the mean of \(R\), as already discussed before since \(\text{R}=1-e^{-\text{WS}(\alpha)}\) should be computed as follows:  
\[\text{E}\left[R(W,S)\big{|}\mathbf{D}\right]=1-\int\limits_{0}^{\infty}\int \limits_{0}^{\infty}e^{-\text{ws}}q(w,s|\mathbf{D})d\textit{r}ds \tag{46}\]  
However, hints on a simpler way to obtain such an expected value of \(R\) have already been hinted at before and will be shown here in the course of computations, together with numerical or simulation methods, which are always possible to give a better approximation if the above ones are generally satisfactory, but it is remarked that the primary meaning of the above approximations (such as E[R] \(\approx\) 0.0735, and more) is not to serve as approximate expected values of the true means but only as typical values that one should expect from computations, so that the experimenter can choose the typical values of the parameters to be assigned a prior PDF with a better knowledge of its implications.