# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
## 3 Results and Discussion
### Method III  
#### 3.3.1 Development of the Method  
The WJ distribution reaches its maximum value at the point \(x=x_{m}\) (\(x_{m}=\
u-\ln(\alpha/\beta)/\beta\)[13]. Thus, it is a natural choice to expand the part \(e^{-\beta(x-\
u)}\) in Equation (2) around \(x=x_{m}\) on the basis of the Taylor formula, overtly,  
\[e^{-\beta(x-\
u)}=\sum_{n=0}^{\infty}\frac{(-1)^{n}\alpha\beta^{n-1}(x-x_{m}) ^{n}}{n!},\]  
resulting in the following expression for the WJ distribution  
\[f(x,\alpha,\beta,\
u)=\frac{\beta}{\Gamma(\alpha/\beta)}e^{-\alpha(x-\
u)- \sum_{n=0}^{\infty}\frac{(-1)^{n}\alpha\beta^{n-1}(x-x_{m})^{n}}{n!}} \tag{21}\]  
Supposed that the truncation of the terms after \(x^{2}\) is reasonable when the value of \(\beta(x-x_{m})\) is sufficiently small, we retain the first three terms in the way of  
\[\sum_{n=0}^{\infty}\frac{(-1)^{n}\alpha\beta^{n-1}(x-x_{m})^{n}}{n!}=\frac{ \alpha}{\beta}-\alpha(x-x_{m})+\frac{\alpha\beta(x-x_{m})^{2}}{2}+\text{O}\left( (x-x_{m})^{3}\right)\]  
and attain the approximate expression for the WJ function  
\[f_{\text{III}}(x,\alpha,\beta,\
u)=\frac{\beta}{\Gamma(\alpha/\beta)}e^{- \frac{\alpha\beta(x-x_{m})^{2}}{2}-\frac{\alpha}{\beta}+\frac{\alpha}{\beta} \ln\frac{\alpha}{\beta}}. \tag{22}\]  
Equation (22) is readily rearranged to the form of a Gaussian function as  
\[f_{\text{III}}(x,\alpha,\beta,\
u)=\frac{\beta e^{\frac{\alpha}{\beta}\ln \frac{\alpha}{\beta}-\frac{\alpha}{\beta}}}{\Gamma(\alpha/\beta)}e^{-\frac{ \alpha\beta(x-x_{m})^{2}}{2}} \tag{23}\]Associating the equation with the formulation of the Gaussian distribution function in Equation (1a), we have two possibilities for assessing the variance \(\sigma\): argument of the exponential-based and pre-exponential factor-based, viz.,  
\[\hat{\sigma}_{\text{III1}}=\sqrt{\frac{1}{\alpha\beta}} \tag{24}\]  
and  
\[\hat{\sigma}_{\text{III2}}=\frac{\Gamma(\alpha/\beta)}{\sqrt{2\pi}\beta e^{ \frac{\alpha}{\beta}\ln\frac{\alpha}{\beta}-\frac{\alpha}{\beta}}} \tag{25}\]  
Concerning the expectation \(\mu_{GS}\), we acquire its estimation of \(\hat{\mu}_{\text{III}}\),  
\[\hat{\mu}_{\text{III}}=x_{m} \tag{26}\]  
or given in the full formula  
\[\hat{\mu}_{\text{III}}=\mu_{\text{WJ}}+\frac{1}{\beta}\Psi\left(\frac{\alpha}{ \beta}\right)-\frac{1}{\beta}\ln\frac{\alpha}{\beta} \tag{27}\]  
In analog to Method I, two normal distributions \(\text{N}(\hat{\mu}_{\text{III}},\hat{\sigma}_{\text{III1}})\) and \(\text{N}(\hat{\mu}_{\text{III}},\hat{\sigma}_{\text{III2}})\) are ready to formulate the WJ distribution in the form of Gaussian functions (labeled individually as CaseIII1 and CaseIII2), as explicitly described by the expressions of  
\[f_{\text{III1}}(x,\alpha,\beta,\
u)=\sqrt{\frac{\alpha\beta}{2\pi}}e^{-\frac{ \alpha\beta(x-x_{m})^{2}}{2}} \tag{28}\]  
for \(\text{N}(\hat{\mu}_{\text{III}},\hat{\sigma}_{\text{III1}})\) and  
\[f_{\text{III2}}(x,\alpha,\beta,\
u)=\frac{\beta e^{\frac{\alpha}{\beta}\ln \frac{\alpha}{\beta}-\frac{\alpha}{\beta}}}{\Gamma(\alpha/\beta)}e^{-\pi( \frac{\alpha\beta\ln\frac{\alpha}{\beta}-\frac{\alpha}{\beta}}{\Gamma(\alpha /\beta)})^{2}(x-x_{m})^{2}} \tag{29}\]  
for \(\text{N}(\hat{\mu}_{\text{III}},\hat{\sigma}_{\text{III2}})\).