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Open AccessArticle

Determining Distribution for the Product of Random Variables by Using Copulas

by Sel Ly 1, Kim-Hung Pho 1,*, Sal Ly 1 and Wing-Keung Wong 2,3,4
1
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 756636, Vietnam
2
Department of Finance, Fintech Center, and Big Data Research Center, Asia University, Tai Chung 41354, Taiwan
3
Department of Medical Research, China Medical University Hospital, Taichung 40402, Taiwan
4
Department of Economics and Finance, Hang Seng University of Hong Kong, Shatin 999077, Hong Kong
*
Author to whom correspondence should be addressed.
Risks 2019, 7(1), 23; https://doi.org/10.3390/risks7010023
Received: 14 January 2019 / Revised: 18 February 2019 / Accepted: 19 February 2019 / Published: 25 February 2019
(This article belongs to the Special Issue Measuring and Modelling Financial Risk and Derivatives)
Determining distributions of the functions of random variables is one of the most important problems in statistics and applied mathematics because distributions of functions have wide range of applications in numerous areas in economics, finance, risk management, science, and others. However, most studies only focus on the distribution of independent variables or focus on some common distributions such as multivariate normal joint distributions for the functions of dependent random variables. To bridge the gap in the literature, in this paper, we first derive the general formulas to determine both density and distribution of the product for two or more random variables via copulas to capture the dependence structures among the variables. We then propose an approach combining Monte Carlo algorithm, graphical approach, and numerical analysis to efficiently estimate both density and distribution. We illustrate our approach by examining the shapes and behaviors of both density and distribution of the product for two log-normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas, and estimate some common measures including Kendall’s coefficient, mean, median, standard deviation, skewness, and kurtosis for the distributions. We found that different types of copulas affect the behavior of distributions differently. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. Our results are the foundation of any further study that relies on the density and cumulative probability functions of product for two or more random variables. Thus, the theory developed in this paper is useful for academics, practitioners, and policy makers. View Full-Text
Keywords: copulas; dependence structures; product of random variables; density functions; distribution functions copulas; dependence structures; product of random variables; density functions; distribution functions
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Ly, S.; Pho, K.-H.; Ly, S.; Wong, W.-K. Determining Distribution for the Product of Random Variables by Using Copulas. Risks 2019, 7, 23.

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