# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
## 3 Results and Discussion
### Method III
#### 3.3.2 Numerical Verifications: As an Accurate, Self-Consistent Approach  
As in the precedent undertakings, we aim to apply the standard normal distribution and its fit parameter set by the WJ distribution function, substituting the numbers of \(\alpha_{0}=12.52,\ \beta_{0}=0.07915\) and \(\upsilon_{0}=63.96\) into the relevant equations for analysis. Figure 6a sums up the plots for the standard normal distribution, \(\text{N}(\hat{\mu}_{\text{III}},\hat{\sigma}_{\text{III1}})\) and \(\text{N}(\hat{\mu}_{\text{III}},\hat{\sigma}_{\text{III2}})\), showing their excellent superposition in the curves. The observation is further validated by their corresponding residuals \(\Delta_{\text{III1}}\) and \(\Delta_{\text{III2}}\) in Figure 6b. Having been used in the calculations, the estimated values of \(\hat{\sigma}_{\text{III1}}=1.0046,\ \ \hat{\sigma}_{\text{III2}}=1.0051\) and \(\hat{\mu}_{\text{III}}=-0.01647\) are obtainable by substituting \(\alpha_{0}\), \(\beta_{0}\) and \(\upsilon_{0}\) into Equations (24)-(26). They are basically indistinguishable from the outcomes of Method I (revealing tiny improvement if not negligible in \(\hat{\sigma}_{\text{III1}}\) and \(\hat{\mu}_{\text{III}}\) over \(\hat{\sigma}_{\text{II}}\) and \(\hat{\mu}_{\text{I}}\), respectively). Similar to the findings in Method I, both \(\hat{\sigma}_{\text{III1}}\) and \(\ \hat{\sigma}_{\text{III2}}\) strikingly are nearly equal and effectively one (with the mean value \(\hat{\mu}_{\text{III}}\) adjacent to \(0\)), signifying the features of the standard normal distribution function (\(\sigma=1\) and \(\mu_{GS}=0\)). Hence, we may draw the same conclusion as in Method I that the transformation of the WJ distribution to the Gaussian function by Method III is both accurate and self-consistent. Figure 6: Analyses of expanding the WJ function by Method III performed at the mode \(x=x_{m}\). (**a**) Gaussian, Case III1 and Case III2 for the standard normal distribution, \(\mathrm{N}(\mu_{\mathrm{III}},\phi_{\mathrm{III}1})\) and \(\mathrm{N}(\mu_{\mathrm{III}},\phi_{\mathrm{III}2})\), correspondingly; (**b**) \(\Delta_{\mathrm{III}1}\) and \(\Delta_{\mathrm{III}2}\) for the residual scatters of \(\mathrm{N}(\mu_{\mathrm{III}},\phi_{\mathrm{III}1})\) and \(\mathrm{N}(\mu_{\mathrm{III}},\phi_{\mathrm{III}2})\) versus \(\mathrm{N}(0,\,1)\), respectively; (**c**) Gaussian, CaseIII1 and CaseIII2 for the cumulative probability distribution functions of \(\mathrm{N}(0,\,1)\), \(\mathrm{N}(\mu_{\mathrm{III}},\phi_{\mathrm{III}1})\) and \(\mathrm{N}(\mu_{\mathrm{III}},\phi_{\mathrm{III}2})\), separately; (**d**) CaseIII1 and CaseIII2 for the respective residual plots of the cumulative probability distribution functions of \(\mathrm{N}(\mu_{\mathrm{III}},\phi_{\mathrm{III}1})\) and \(\mathrm{N}(\mu_{\mathrm{III}},\phi_{\mathrm{III}2})\) relative to \(\mathrm{N}(0,\,1)\); (**e**) \(\sigma_{\mathrm{III}1}\) and \(\sigma_{\mathrm{III}2}\) for relationship between \(\alpha\) and \(\beta\) when \(\hat{\sigma}_{\mathrm{III}1}=1\) and \(\hat{\sigma}_{\mathrm{III}2}=1\), respectively; (**f**) Relationship between \(\alpha\) and \(\beta\) assuming \(\hat{\mu}_{\mathrm{III}}=0\) with \(\mu_{WJ}\in(0.05,\,0.1,\,0.2,0.5)\) according to Equation (27), individually. Consistent with the well-defined observation above, Figure 6c evaluates the accuracy of the cumulative probability distribution functions for \(\mathrm{N}(0,\,1)\), \(\mathrm{N}(\hat{\mu}_{\mathrm{III}},\hat{\sigma}_{\mathrm{III}})\) and \(\mathrm{N}(\hat{\mu}_{\mathrm{III}},\hat{\sigma}_{\mathrm{III}2})\), with their residuals, plotted in Figure 6d. The relationships of the parameter spaces are exemplified for \(\hat{\sigma}_{\mathrm{III}}=1\) and \(\hat{\sigma}_{\mathrm{III}2}=1\) in Figure 6e, which are only a function of \(\alpha\) and \(\beta\) and show the monotonic decrease in \(\beta\) with increasing \(\alpha\), separately. In contrast, the estimation \(\hat{\mu}_{\mathrm{III}}\) is a function of \(\mu_{WI}\) in addition to \(\alpha\) and \(\beta\), as delineated in Equation (27).