## 5 Methods
### Probability Distributions  
A random variable \(X\), here the wind speed, is said to have the 2-parameter Weibull distribution \(W(a,b)\) with parameters \(a>0\), \(b>0\) if its probability density function (PDF) for \(x\geq 0\) is given by,  
\[f(x)=\frac{a}{b^{a}}x^{a-1}\exp\left(-\left(\begin{array}{c}\frac{x}{b}\\ \end{array}\right)^{a}\right) \tag{1}\]  
and cumulative distribution function (CDF) is given by,  
\[F(x)=1-\exp\left(-\left(\begin{array}{c}\frac{x}{b}\\ \end{array}\right)^{a}\right) \tag{2}\]  
where \(a\) is the dimensionless shape parameter and \(b\) is the scale parameter in units of the wind speed, here in m/s. The parameter \(a\) specifies the shape of the Weibull distribution, which sets the width of the wind speed data distribution. The shape factor determines the consistency of the wind speed at a particular location. It is known that, generally, \(a\) varies from 1.5 to 3 for most wind speed conditions in the world [58; 59]. Small values of \(a\) signify very variable winds, while less variable winds are characterised by higher values of \(a\). The scale parameter \(b\) determines the abscissa scale of the wind speed data distribution. It is proportional to the mean wind speed, with higher values for the locations with strong wind and lower values for still locations.  
A random variable \(X\) is said to have the 3-parameter Weibull distribution \(W(a,b,c)\) with parameters \(a>0\), \(b>0\), \(c\geq 0\) if its PDF for \(x\geq c\) is given by,  
\[f(x)=\frac{a}{b^{a}}(x-c)^{a-1}\exp\left(-\left(\begin{array}{c}\frac{x-c}{b }\\ \end{array}\right)^{a}\right) \tag{3}\]  
and its CDF is given by,  
\[F(x)=1-\exp\left(-\left(\begin{array}{c}\frac{x-c}{b}\\ \end{array}\right)^{a}\right) \tag{4}\]  
where \(a\) is the dimensionless shape parameter, \(b\) is the scale parameter, and \(c\) is the location parameter.  
The exponentiated distribution is obtained by raising the base CDF to the positive parameter \(\gamma\). A random variable \(X\) is said to have the exponentiated distribution if its CDF \(F(x)\) and PDF \(f(x)\) are given by,  
\[F(x)=\left[\begin{array}{c}G(x)\\ \end{array}\right]^{\gamma}, \tag{5}\]  
\[f(x)=\gamma[G(x)]^{\gamma-1}g(x) \tag{6}\]  
where \(\gamma>0\), and \(G(x)\) and \(g(x)\) are CDF and PDF of the base distribution, respectively. [PERSON] and [PERSON] in [39] applied exponentiation to the 2-parameter Weibull distribution to obtain the exponentiated Weibull distribution. A random variable \(X\) is said to have the exponentiated Weibull distribution \(EW(a,\ b,\gamma)\) with parameters \(a>0\), \(b>0,\ \gamma>0\) if its PDF for \(x>0\) is given by,  
\[f(x)=\frac{\gamma a}{b}\Big{(}\frac{x}{b}\Big{)}^{a-1}\Big{[}1-\exp\Big{(}- \Big{(}\frac{x}{b}\Big{)}^{a}\ \Big{)}^{\gamma-1}\exp\Big{(}-\Big{(}\frac{x}{b}\Big{)}^{a}\Big{)} \tag{7}\]  
and its CDF is given by,  
\[F(x)=\Big{[}1-\exp\Big{(}-\Big{(}\frac{x}{b}\Big{)}^{a}\ \Big{)}\Big{]}^{\gamma}. \tag{8}\]  
Here both \(a\), \(\gamma\) are shape parameters and \(b\) is a scale parameter. For \(\gamma=1\), we obtain the 2-parameter Weibull distribution. The more detailed explanation and further information for the EW distribution can be found in [39; 42; 45].  
In further text, we will use the following abbreviations for the probability distributions: 2-parameter Weibull--W2, 3-parameter Weibull--W3, and exponentiated Weibull--EW.