# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
## 3 Results and Discussion
### Method 1  
#### 3.1.1 Development of the Method  
We first expand the term of \(e^{-\beta x}\) in the exponential part \(e^{-\beta(x-\
u)}\) in Equation (2) on the basis of the Taylor formula around the point \(x=0\), explicitly,  
\[e^{-\beta x}=\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}\beta^{n}x^{n}}{n!},\]  
then carry out the substitution, giving  
\[f(x,\alpha,\beta,\
u)=\frac{\beta}{\Gamma(\alpha/\beta)}e^{-\alpha(x-\
u)-e^ {\beta\
u}\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}\beta^{n}x^{n}}{n!}} \tag{4}\]  
Considering the truncation of the terms after \(x^{2}\) when the value of \(\beta x\) is small enough, we take the first three terms in the sum  
\[\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}\beta^{n}x^{n}}{n!}=1-\beta x+ \frac{\beta^{2}x^{2}}{2}+O\Big{(}\beta^{3}x^{3}\Big{)}\]  
and acquire the approximate expression for the WJ function in the form of  
\[f_{1}(x,\alpha,\beta,\
u)=\frac{\beta}{\Gamma(\alpha/\beta)}e^{-\frac{\beta^{ n}\beta^{n}x^{2}}{2}-\left(\alpha-\beta^{n}\right)x+\alpha\
u-e^{\beta\
u}} \tag{5}\]  
The exponential part of Equation (5) is subsequently rearranged to the square form about \(x\) (i.e., the exponential term of a Gaussian function) as  
\[f_{1}(x,\alpha,\beta,\
u)=\frac{\beta e^{\frac{(\beta^{n}\beta^{n}-\alpha^{2} )}{2^{n}\beta^{n}\beta^{2}}+\alpha\
u-e^{\beta\
u}}}{\Gamma(\alpha/\beta)}e^{ -\frac{\beta^{n}\beta^{n}(\alpha-\beta^{n}-\alpha^{2})}{2}} \tag{6}\]  
As Equation (6) is the result of truncating the Taylor expansion to second order, however, the expression is not certainly normalized as a probability distribution. To guarantee the normalization and still a good approximation to a Gaussian function, we adopt the line of rationalization: Comparing the equation with the formulation of the Gaussian function in Equation (1a), we have two options for the estimation of \(\sigma\), exponential based and prefactor based, that is,  
\[\vartheta_{\rm I1}=\sqrt{\frac{1}{\beta^{2}e^{\beta\
u}}} \tag{7}\]  
and  
\[\vartheta_{\rm I2}=\frac{\Gamma(\alpha/\beta)}{\sqrt{2\pi}\beta e^{\frac{ \beta^{2}}{2^{n}\beta^{2}}-\frac{\beta}{2}+\alpha\
u-\frac{\beta^{n}}{2}}} \tag{8}\]With regards to the expectation \(\mu_{GS}\), we have its estimation of \(\hat{\mu}_{\rm I}\),  
\[\hat{\mu}_{\rm I}=\frac{\beta e^{\beta\
u}-\alpha}{\beta^{2}e^{\beta\
u}} \tag{9}\]  
Therefore, we have come to the two equations, (7) and (8), for estimating \(\sigma\) on the basis of the parameters of the WJ distribution function. In combination with Equation (9), two normal distributions \(\rm N(\hat{\mu}_{\rm I},\hat{\sigma}_{\rm I1})\) and \(\rm N(\hat{\mu}_{\rm I},\hat{\sigma}_{\rm I2})\) are available to approximate the WJ distribution to be expressed as Gaussian functions, labeled respectively as Casell1 and Casel2 for the convenience of discussion. In explicit, the expressions representing the normal distributions are  
\[f_{\rm I1}(x,\alpha,\beta,\
u)=\frac{\beta e^{\beta\
u/2}}{\sqrt{2\pi}}e^{- \frac{\beta^{2}e^{\beta\
u}(x-\frac{\beta^{2}}{2}e^{\beta\
u})^{2}}{2}} \tag{10}\]  
for \(\rm N(\hat{\mu}_{\rm I},\hat{\sigma}_{\rm I1})\) and  
\[f_{\rm I2}(x,\alpha,\beta,\
u)=\frac{\beta e^{\frac{(\beta e^{\beta\
u}- \alpha^{2})^{2}+\alpha\
u-\beta\
u}{2\beta^{2}\beta^{2}}}e^{-\pi(\frac{\beta e ^{2}e^{\beta\
u}-\frac{\beta}{2}+\alpha\
u-\frac{\beta e^{\beta\
u}}{2}})}}{ \Gamma(\alpha/\beta)}e^{-\pi(\frac{\beta e^{2}e^{\beta\
u}-\frac{\beta}{2}+ \alpha\
u-\frac{\beta e^{\beta\
u}}{2}}{\Gamma(\frac{\beta}{\beta})})}\ (x-\frac{(\beta e^{\beta\
u}- \frac{\beta}{2})^{2}}{\beta^{2}e^{\beta\
u}})^{2} \tag{11}\]  
for \(\rm N(\hat{\mu}_{\rm I},\hat{\sigma}_{\rm I2})\).