**3. Copulas**

In this section, we will briefly discuss the copula theory related to the theory developed in our paper. Readers may refer to (Cherubini et al. 2004; Joe 1997; Nelsen 2007; Tran et al. 2015, 2017) for more information. Let I = [0, 1] be the closed unit interval and I2 = [0, 1] × [0, 1] be the closed unit square interval. We first state the most basic definition of copula in two dimensions in the following:

**Definition 1.** *(Copula) A 2-copula (two-dimensional copula) is a function C:* I2 → I *satisfying the following conditions:*


*(iii) for any u*1, *u*2, *v*1, *v*2 ∈ I *with u*1 ≤ *u*2 *and v*1 ≤ *v*2,

$$
\mathbb{C}(\boldsymbol{\mu}\_2, \boldsymbol{\upsilon}\_2) + \mathbb{C}(\boldsymbol{\mu}\_1, \boldsymbol{\upsilon}\_1) - \mathbb{C}(\boldsymbol{\mu}\_2, \boldsymbol{\upsilon}\_1) - \mathbb{C}(\boldsymbol{\mu}\_1, \boldsymbol{\upsilon}\_2) \ge 0.
$$

In copula theory, Sklar proposed a very important theorem in 1959 called Sklar's Theorem (Cherubini et al. (2004); Joe (1997); Nelsen (2007)), which plays the most important role in this theory. It tells us that given a random vector (*<sup>X</sup>*1, *<sup>X</sup>*2) with absolutely continuous marginal distribution functions *FX*1 and *FX*2 , respectively, and its joint distribution function denoted by *H*, and then there exists a unique copula *C* such that

$$\begin{array}{rcl}H(\mathbf{x}\_{1},\mathbf{x}\_{2})&=&\mathcal{C}\left(F\_{\mathbf{X}\_{1}}(\mathbf{x}\_{1}),F\_{\mathbf{X}\_{2}}(\mathbf{x}\_{2})\right),\\h(\mathbf{x}\_{1},\mathbf{x}\_{2})&=&\frac{\partial^{2}}{\partial \mathbf{x}\_{1}\partial \mathbf{x}\_{2}}H(\mathbf{x}\_{1},\mathbf{x}\_{2})=\mathcal{C}\left(F\_{\mathbf{X}\_{1}}(\mathbf{x}\_{1}),F\_{\mathbf{X}\_{2}}(\mathbf{x}\_{2})f\_{\mathbf{X}\_{1}}(\mathbf{x}\_{1})f\_{\mathbf{X}\_{2}}(\mathbf{x}\_{2})\right),\end{array} \tag{5}$$

where *<sup>c</sup>*(*<sup>u</sup>*, *v*) := *∂*2 *<sup>∂</sup>u∂v<sup>C</sup>*(*<sup>u</sup>*, *v*) denotes density of copula *C*, *fXi* is probability density function (PDF) of *Xi*, *i* = 1, 2, and *h*(*<sup>x</sup>*1, *<sup>x</sup>*2) is the joint density function of *X*1 and *X*2. Copula is used to combine several univariate distributions together into bivariate [multivariate] settings so as the copula *C* can capture the dependence structure of (*<sup>X</sup>*1, *<sup>X</sup>*2) [(*<sup>X</sup>*1, ··· , *Xn*)]. For any copula *C*, we have the bounds

$$\mathcal{W}(\mathfrak{u}, \upsilon) \le \mathcal{C}(\mathfrak{u}, \upsilon) \le \mathcal{M}(\mathfrak{u}, \upsilon)\_{\prime\prime}$$

where the copula *<sup>W</sup>*(*<sup>u</sup>*, *v*) := max(*u* + *v* − 1, 0) captures counter-monotonicity structure; that is, *X*2 = *f*(*<sup>X</sup>*1) a.s., where *f* is strictly decreasing, while the copula *<sup>M</sup>*(*<sup>u</sup>*, *v*) := min(*<sup>u</sup>*, *v*) is used to capture comonotonicity; that is, *X*2 = *f*(*<sup>X</sup>*1) a.s., where *f* is strictly increasing. In case *X*1 and *X*2 are independent, they follow copula denoted by <sup>Π</sup>(*<sup>u</sup>*, *v*) := *uv*. Copulas can be used not only to model the dependence structure of the variables, but also capture the correlation between the variables. The Kendall's coefficient *τ* can be expressed in terms of copulas as shown in the following:

$$\pi(X\_1, X\_2) = \pi(\mathbb{C}) = 4 \int \int\_{\mathbb{T}^2} \mathbb{C}(u, v) d\mathbb{C}(u, v) - 1. \tag{6}$$

In the next section, we will derive the two main propositions regarding formulas that can be used to determine the probability density and probability distribution of the quotient of dependent random variables by using copulas. In addition, we will apply the results to derive some corollaries on PDFs, CDFs, and median of the ratios in case *X*1 and *X*2 are normal distributed and they follow the Gaussian copulas.
