In general, suppose $G_i(u_i) \in [0, 1]$ is a univariate, uniform distribution with $u_i = u_1, \dots u_n$ and $i \in \mathbb{N}$ ($i$ is an element of set $\mathbb{N}$). Then, we define a copula function as follows:

$$C[G_1(u_1), \dots, G_n(u_n)] = F_n[F_1^{-1}(G_1(u_1)), \dots, F_n^{-1}(G_n(u_n)); \rho_F]$$

where:

• $G_i(u_i)$ are the marginal distributions that have no well-known properties
• $F_n$ is the joint cumulative distribution function
• $F_i^{-1}$ is the inverse of $F_i$
• $\rho_F$ is the correlation structure of $F_n$

Put in words, the above equation reads:

Given the marginal distributions $G_1(u_1)$ to $G_n(u_n)$, there exists a copula function that allows the mapping of the marginal distributions $G_1(u_1)$ to $G_n(u_n)$ via $F^{-1}$ and the joining of the (abscise values) $F_i^{-1}(G_i(u_i))$ to a single, $n$-variate function $F_n[F_1^{-1}(G_1(u_1)), \dots, F_n^{-1}(G_n(u_n))]$ that has a correlation structure of $\rho_F$.