**6. Conclusions**

Determining distributions of the functions of random variables is a very crucial task and this problem has attracted a number of researchers because there are numerous applications in Economics, Science, and many other areas, especially in the areas of finance including risk managemen<sup>t</sup> and option pricing. However, to the best of our knowledge, the problem of determining distribution functions of quotient of dependent random variables using copulas has not been widely studied and, as far as we know, no published paper or working paper has done the work we are doing in this paper. Thus, to bridge the gap in the literature, in this paper, we first develop two general propositions on both density and distribution functions for the quotient *Y* = *X*1 *X*2 and the ratio of one variable over the sum of two variables *Z* := *X*1 *X*1+*X*2 of two dependent random variables *X*1 and *X*2 by using copulas. We then derive two corollaries on both density and distribution functions for the two quotients *Y* = *X*1 *X*2 and *Z* = *X*1 *X*1+*X*2 of two dependent normal random variables *X*1 and *X*2 in case of Gaussian Copulas by applying the two main general propositions developed in our paper. From the results, we derive the corollaries on the median for the ratios of both *Y* and *Z* of two normal random variables *X*1 and *X*2. Furthermore, the result of median for *Z* is also extended to a larger family of symmetric distributions and symmetric copulas of *X*1 and *X*2.

Since the density and the CDF formula of the ratios of both *Y* and *Z* are in terms of integrals and are very complicated, we cannot obtain the exact forms of the density and the CDF. To circumvent the difficulty, in this paper, we propose to use the Monte Carlo algorithm, numerical analysis, and graphical approach that can efficiently compute complicated integrals and study the behaviors of both density and distribution and the changes of their shapes when parameters are changing. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-*t*, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts on behavior of distributions, and since Gaussian and Student-*t* Copulas belong to an elliptical family, they similarly act on shapes of *Y* and *Z* in the same fashion. We also document the effects when using Archimedean copulas including Clayton, Gumbel, Frank, and Joe Copulas. However, there are also some differences, especially on location and scale effects. For example, distribution of *Z* does not change the median and its shape is always symmetric for all investigated copulas while the random variable *Y* is affected in skewness, median and spread. Our findings are useful for academics in their study of the shapes, center and spread of both density and distribution functions for the ratios by using different copulas. Since the ratios in different copulas are widely used in many important empirical studies in Finance, Economics, and many other areas, our findings are useful to practitioners in Finance, Economics, and many other areas who need to study the shapes, center and spread of the ratios by using different copulas in their analysis and useful to policy makers if they need to consider the shapes, center and spread of both density and distribution functions for the ratios by using different copulas in their policy decision-making. Finally, we note that, although all of the propositions and corollaries developed in our paper are relatively easy to derive, all the results developed in our paper are useful to both academics and practitioners because there is a wide range of applications with variables that have a negative range. Readers may refer to (Chang et al. 2016, 2018a, 2018b, 2018c; Chang et al. 2015; Wong 2016) for more information on the applications in different areas.

**Author Contributions:** Writing—original draft preparation, S.L. (Sel Ly) and K.-H.P.; writing—review and editing,W.-K.W.; visualization, S.L. (Sel Ly) and S.L. (Sal Ly).

**Funding:** This research has been supported by Ton Duc Thang University, Asia University, China Medical University Hospital, Hang Seng University of Hong Kong, Research Grants Council (RGC) of Hong Kong (project number 12500915), and Ministry of Science and Technology (MOST, Project Numbers 106-2410-H-468-002 and 107-2410-H-468 -002-MY3), Taiwan.

**Acknowledgments:** The authors are grateful to Michael McAleer, the Editor-in-Chief, and anonymous referees for substantive comments that have significantly improved this manuscript. The fourth author would like to thank Robert B. Miller and Howard E. Thompson for their continuous guidance and encouragement.

**Conflicts of Interest:** The authors declare no conflict of interest.
