*5.1. Gaussian Copulas*

We first investigate dependence structures of *X*1 and *X*2 through Gaussian Copulas *Cr*(*<sup>u</sup>*, *v*) and observe the shapes of the corresponding distributions for both *Y* and *Z*.

$$\mathbb{C}\_{r}(u,v) = \frac{1}{2\pi\sqrt{1-r^2}} \int\_{-\infty}^{\Phi^{-1}(u)} \int\_{-\infty}^{\Phi^{-1}(v)} \exp\left(-\frac{s^2 - 2rst + t^2}{2(1-r^2)}\right) ds dt,\tag{46}$$

where <sup>Φ</sup>−<sup>1</sup>(*x*) is the inverse of standard normal CDF and *r* is Pearson's correlation coefficient between *X*1 and *X*2, |*r*| < 1. We now consider the cases with *r* = −0.9, −0.5, 0, 0.5, 0.9. When *r* = 0, it is corresponding to the independence situation, and we ge<sup>t</sup> PDFs and CDFs of *Y* and *Z* shown in Figures 1 and 2, respectively. As can be seen from the Figures and Tables 1 and 2, when the parameter *r* varies from negative to positive, the median is totally equal to the Pearson's correlation coefficient *r*. The more correlated, that is, the higher |*r*| between *X*1 and *X*2 is, the smaller the spread, that is, IQR of *Y*, becomes. In contrast to *Y*, the center of *Z* is definitely unchanged (0.5), but the scale parameter of *Z* is smaller indicated by the higher height of the density. The shapes of both *Y* and *Z* are symmetric.

**Table 1.** Some percentiles of *Y* = *X*1/*X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Gaussian Copulas.


**Table 2.** Some percentiles of *Z* = *<sup>X</sup>*1/(*<sup>X</sup>*1 + *<sup>X</sup>*2), where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Gaussian Copulas.


<sup>2</sup> We would like to show our appreciation to the anonymous reviewer to giving us helpful comments so that we could draw this conclusion.

**Figure 1.** PDFs and CDFs of the ratio *Y* = *X*1 *X*2 , where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Gaussian Copulas.

**Figure 2.** PDFs and CDFs of the ratio *Z* = *X*1 *X*1+*X*2 , where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Gaussian Copulas.
