# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
## 3 Results and Discussion
### On Mutual Fitting of WJ Distributions and Arbitrary Gaussian Functions  
Having performed the analysis of the relationship between the WJ distribution and the standard Gaussian function above, we now go over the capacity of the WJ distribution to fit a general Gaussian function, or vice versa, and the merits of fitting general WJ distributions by Gaussian functions. Figure 7a lists several selected examples of fitting the Gaussian function, \(\mathrm{N}(0,\,1)\), \(\mathrm{N}(-1,\,0.5)\), \(\mathrm{N}(1,\,2)\), and \(\mathrm{N}(2,\,5)\), by the WJ distribution (for convenience of later discussion, the subscripts \(0,\,1,\,2\) and \(3\) are assigned in order from left to right). Table 1 sums up the parameter values of the designated curves and the corresponding fitting outcomes as obtained. In the figure, each individual fitting result is in excellent agreement (Gaussian functions in solid dots and fitting WJ distributions in solid lines), as statistically reflected in their correlation coefficients of almost one (0.9997 or larger for all the examples in actual). Based on the obtained parameters in Table 1, we compute the value of \(\mu_{WI}\) in terms of \(\
u\) and obtain \(\mu_{WIJ0}=0.0235\), \(\mu_{WI1}=-0.9835\), \(\mu_{WI2}=1.0104\) and \(\mu_{WI3}=2.1504\) accordingly. A more direct calculation produces the ratios of  
\[\frac{\alpha_{0}}{\alpha_{1}}\approx 0.5,\,\,\frac{\alpha_{0}}{\alpha_{2}} \approx 2\,\,\mathrm{and}\,\,\frac{\alpha_{0}}{\alpha_{3}}\approx 4.96\]  
as well as  
\[\frac{\beta_{0}}{\beta_{1}}\approx 0.5,\,\,\frac{\beta_{0}}{\beta_{2}}\approx 1.9 \,\,\mathrm{and}\,\,\frac{\beta_{0}}{\beta_{3}}\approx 4.99,\]  
which are satisfactorily consistent with the previous derivations of  
\[\alpha^{*}=\frac{\alpha}{\sigma}\,\,\mathrm{and}\,\,\beta^{*}=\frac{\beta}{ \sigma}.\]  
Still, setting \(\delta={\mu_{WI}}^{*}-\left(\mu_{GS}+\sigma\mu_{WI}\right)\) gives the results of \(\delta_{1}\approx 0.0048\), \(\delta_{2}\approx-0.0366\) and \(\delta_{3}=0.0329\), all of which are close to \(0\) as expected. Obviously, the sound fits of the Gaussian functions by the WJ distributions sanction the validity of the rescaling scheme as deduced in the section of Method I. The scrutiny in the previous sections has attested that the WJ distribution is capable to well characterize the Gaussian function of arbitrary parameters. Next, we investigate the effect of fitting the WJ distribution by the Gaussian function. Figure 7b shows four cases of Gaussian functions fitting to WJ distributions, and Table 1 summarizes the parameters taken for the WJ distributions and the fitting results from the Gaussian functions. As indicated in the first panel of Figure 7b (WJ distribution in hollow squares and fitting Gaussian function in solid line), the fit is expectedly desired since the WJ function takes the specific set of the parameters \(\alpha_{0}=12.52\), \(\beta_{0}=0.07915\) and \(\
u_{0}=63.96\). Combined with the preceding analysis (the WJ distribution fits almost the standard normal function), the WJ distribution and the Gaussian function can mutually represent each other with these particular parameters. This phenomenon, nonetheless, alters dependent on the set of the parameters specified when we vary the values of \(\alpha\), \(\beta\) and \(\
u\). It is true that the second plot in Figure 7b shows a significantly unsatisfactory fit in comparison to the first, while the Gaussian functions in the third and fourth figures no longer portray properly the trends of the corresponding WJ distributions, which are enormously distinct by the difference in the WJ distribution.