# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
## 3 Results and Discussion
### Method 1
#### 3.1.4 Investigation of Parameter Solution Spaces of the WJ Distribution  
It is known that the standard deviation \(\sigma\) of a Gaussian function determines the shape of its curve, while the shape of the WJ distribution is jointly controlled by both parameters \(\alpha\) and \(\beta\). Thus, it is significant to study the interrelation between the estimate \(\sigma\) with respect to the solution space of \(\alpha\) and \(\beta\). Inspecting Equations (7) and (8), we find that the estimate \(\sigma\) is related to \(\
u\) in addition to the parameters \(\alpha\) and \(\beta\). Thus, we need to initially derive the expression of \(\
u\) in relation to \(\alpha\) and \(\beta\) in terms of the maximal value of the distribution function as the starting point of calculation. When the WJ distribution can well represent the Gaussian function, actually, the corresponding curve reflects a certain degree of axial symmetry, indicating that the extreme point should approach \(x=\mu_{GS}\). Since the mode of the WJ distribution is located at the point  
\[x=\
u-\frac{1}{\beta}ln\frac{\alpha}{\beta},\]  
it leads to  
\[\
u=\mu_{GS}+\frac{1}{\beta}ln(\frac{\alpha}{\beta}). \tag{13}\]Substituting Equation (13) into Equations (7) and (8), we deduce the following expressions for \(\hat{\sigma}_{\text{II}}\) and \(\hat{\sigma}_{\text{I2}}\) (noted as \(D_{\text{II}}\) and \(D_{\text{I2}}\) for the specific selection of \(\
u\)), respectively,  
\[D_{\text{II}}=\sqrt{\frac{1}{\alpha\beta e^{\beta\hat{\eta}_{GS}}}} \tag{14}\]  
and  
\[D_{\text{I2}}=\frac{\Gamma(\alpha/\beta)}{\sqrt{2\pi}\beta e^{\frac{\beta \left(\frac{1-2\beta\beta_{GS}}{2\mu_{GS}}+1n(\beta)-1\right)+\alpha\mu_{GS}}}} \tag{15}\]  
The parameter \(\mu_{GS}\) is identified provided that the Gaussian function is specified. Afterward, we can solve Equations (14) and (15) for \(\alpha\) and \(\beta\) of the WJ function with respect to a given value of \(\mu_{GS}\) by setting \(D_{\text{II}}\) and \(D_{\text{I2}}\) to desired quantities. Figure 3 shows the solution space in the form of \(\alpha\) vs. \(\beta\) with \(D_{\text{II}}\) and \(D_{\text{I2}}\) set to one. In detail, Figure 3a plots the relationship curves when \(\mu_{GS}\) is set to 0, and the values of \(D_{\text{II}}\) and \(D_{\text{I2}}\) are given to one. As shown in the figure, the curves are awfully sensitive to the change of \(\alpha\) when \(\alpha\) is less than one, whereas they become sensitive to the change in \(\beta\) when \(\alpha\) is greater than four. The two curves of \(D_{\text{II}}\) and \(D_{\text{I2}}\) apparently overlap when \(\alpha\) is greater than two, meaning that the same set of the \(\alpha\) and \(\beta\) values is available to make \(D_{\text{II}}=D_{\text{I2}}\) and the WJ function is anticipated to have a better characterization of the Gaussian function. As \(\alpha\) decreases, the two curves grow away from each other so that \(D_{\text{II}}\) and \(D_{\text{I2}}\) do not share the same solution anymore and thus cannot take one at the same time. Instead of \(\mu_{GS}\) by utilizing Equation (13), we can tackle the solution space of \(\hat{\sigma}\) with respect to \(\alpha\) and \(\beta\) by substituting  
\[\
u=\mu_{WJ}+\frac{1}{\beta}\Psi\left(\frac{\alpha}{\beta}\right)\]  
into Equations (7) and (8) according to Equation (3), separately. As an adjustable parameter, \(\mu_{WJ}\) is going to play the role of \(\mu_{GS}\) and can be investigated with suitable values. Upon the condition of \(\hat{\sigma}_{\text{II}}=1\) and \(\hat{\sigma}_{\text{I2}}=1\), the curves in Figure 3b depict the relationship between \(\alpha\) and \(\beta\) for the particular case of \(\mu_{WJ}=0\). The trend of the curve for \(\hat{\sigma}_{\text{II}}\) is analogous to that of Figure 3a, more sensitive to a change in \(\alpha\). Nevertheless, \(\hat{\sigma}_{\text{I2}}\) appears particularly disparate: The curve first rises to the peak and then declines, displaying more sensitivity to a change in \(\beta\).