# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
## 3 Results and Discussion
### Method 1
#### 3.1.2 Numerical Appraisal: As an Accurate, Self-Consistent Method  
As the important special case of the general Gaussian function (1a), we shall first focus on the aspects of analysis based on the standard normal distribution \(x\sim\rm N(0,\,1)\), which can be recovered from the former by the transform of  
\[x=\frac{\
u-u_{GS}}{\sigma}\.\]  
The relevant interval probability is evaluable by the expression  
\[P(a<x<b)=\Phi\bigg{(}\frac{b-\mu_{GS}}{\sigma}\bigg{)}-\Phi\bigg{(}\frac{a- \mu_{GS}}{\sigma}\bigg{)}\]  
(\(\Phi[(x-\mu_{GS})/\sigma)\)) is the cumulative probability function of the normal distribution). Illuminating the relationship between the WJ function and the normal distribution, the assessing procedure proceeds in the following way: Appropriately select a set of discrete points from the standard normal distribution (1b) as an experimental dataset, then fit it with the WJ distribution (2) and thus obtain the corresponding parameters \(\alpha_{0}=12.52\), \(\beta_{0}=0.07915\) and \(v_{0}=63.96\) (\(R^{2}\approx 0.9997\)). As shown in Figure 2a, in which the dots are the Gaussian values, and the curve is the fitting outcome of the WJ function, it is clear that both of them match optimally and are consistent with the almost unity of the correlation coefficient (0.9997). The calculation error is estimated by the scattering points \(\Delta_{\rm I}\) in Figure 2b, with each point close to 0 (the maximum error is less than 0.005). As a result, the WJ distribution proves its excellent capability in characterizing the Gaussian function \(\rm N(0,\,1)\), suggesting that such deliberation may be extendable to general cases as outlined below. There is another way to examine the accuracy of the Taylor expansion for the WJ distribution in the Gaussian form; that is, the fitting parameters as obtained above are substituted into Equations (7)-(9) to generate the estimated parameters of Gaussian functions as \(\hat{\sigma}_{\rm I1}=1.0052,\ \hat{\sigma}_{\rm I2}=1.0051\) and \(\hat{\mu}_{\rm I1}=-0.01648\), correspondingly. The values of \(\hat{\sigma}_{\rm I1}\) and \(\hat{\sigma}_{\rm I2}\) are remarkably almost equal and virtually one, together with the mean value \(\hat{\mu}_{\rm I}\) close to 0. Such findings are in accord with the standard normal distribution function (\(\sigma=1\) and \(u_{GS}=0\)). Therefore, we may draw a conclusion that the transformation of the WJ distribution to the Gaussian function by Method I is both accurate and self-consistent. In Figure 2b, \(\Delta_{\rm II}\) and \(\Delta_{\rm II}\) present the residualsof \(\mathrm{N}(\hat{\mu}_{\mathrm{I}},\hat{\sigma}_{11})\) and \(\mathrm{N}(\hat{\mu}_{\mathrm{I}},\hat{\sigma}_{2})\) compared to \(\mathrm{N}(0,\ 1)\) over the entire curves, respectively. Similar to the precedent estimation, it is apparent that each error point is near zero and less than \(0.01\). Such proceeding prospectively reveals the feasibility of assessing the parameters \(\sigma\) and \(\mu\) of the standard normal distribution according to Equations (7)-(9) with \(\alpha_{0}\), \(\beta_{0}\) and \(v_{0}\) as acquired through the fitting. In brief, the truncation method incurred above is substantiated to be valid in exploiting the relation between the WJ distribution and the Gaussian function. The validity of Method I may be further verified by appraising the related cumulative probability distribution function \(F(x)\). Since we consider in this work the cases such as the standard normal distribution with the meaningful values around the origin exclusively, it stays justified to take the lower limit to be a finite number, say \(-4\), to replace \(-\infty\) in the integral \(F(x)=\int_{-\infty}^{x}f(x)dx\) to compute the numerical solution of the cumulative probability distribution function.