May 18, 2026
Copulas glue together marginal distributions to define joint distributions.
If the random variable X has cumulative distribution function {F(x) = P(X\le x)} then {F^{-1}(X)} is uniformly distributed on the unit interval [0,1].
In the case of two random variables the copula of X and Y is the joint distribution of F^{-1}(X) and G^{-1}(Y) where F and G are the cumulative distributions of X and Y respectively: {C(u,v) = C^{X,Y}(u,v) = P(F^{-1}(X) \le u, G^{-1}(Y) \le v)}.
Exercise: Show C(u,v) = H(F(u),G(v)) where and H is the joint distribution of X and Y and F and G are the marginal distributions of X, and Y respectively.
Exercise: Show H(x,y) = C(F^{-1}(x), G^{-1}(y)).
This shows how to use the copula and marginal distributions to recover the joint distribution.
An equivalent definition is a copula is a probability measure on [0,1]^2 with uniform marginals.
Exercise: Prove this.
If U and V are independent, uniformly distributed random variables on the unit interval then C(u,v) = P(U\le u, V\le v) = uv. This corresponds to uniform measure on the square [0,1]^2.
If V=U then their joint distribution is \begin{aligned} C(u,v) &= P(U\le u, V\le v) \\ &= P(U\le u, U\le v) \\ &= P(U\le \min\{u, v\}) \\ &= \min\{u,v\} \\ &= M(u,v) \end{aligned} This corresponds to uniform measure on the diagonal from (0,0) to (1,1).
If V=1-U then their joint distribution is \begin{aligned} C(u,v) &= P(U\le u, V\le v) \\ &= P(U\le u, 1-U\le v) \\ &= P(1-v\le U\le u) \\ &= \max\{u - (1 -v), 0\} \\ &= \max\{u + v - 1, 0\} \\ &= W(u,v) \end{aligned} This corresponds to uniform measure on the anti-diagonal from (0,1) to (1, 0).
Exercise: (Fréchet-Hoeffding) For every copula, C, W \le C \le M.
Hint: For the upper bound use H(x,y) \le F(x) and H(x,y) \le G(y). For the lower bound note 0\le C(u_1,v_1) - C(u_1, v_2) - C(u_2, v_1) + C(u_2, v_2) for u_1 \ge u_2 and v_1 \ge v_2.
Random variables X and Y are independent if and only if their copula is uniform measure on [0,1]^2. To model random variables that are independent near their means but become perfectly correlated at extreme values we can use, e.g., uniform measure on [a,b] \times [a,b], 0 < a < b < 1, and uniform measure the lines from (0,0) to (a,a) and from (b,b) to (1,1) with weights w_a and w_b such that w_a + (b - a)^2 + w_b = 1. Similarly we could model perfect anti-correlation at extreme values using uniform measure on lines from (a,b) to (0,1) and from (b,a) to (1, 0). (Draw the picture.)
An analytically tractable continuous version is provided by the Plackett copula C(u,v;\theta) = \frac{[1 + (\theta-1)(u + v)] - \sqrt{[1 + (\theta - 1)(u + v)]^2 - 4\theta(\theta - 1)uv}} {2(\theta - 1)}, \theta > 0, \theta\not=1. As \theta\to 1 we have C(u,v;\theta)\to uv.
The partial derivatives \partial C/\partial u and \partial C/\partial v and be inverted in closed form and be used to generate random variables having the Plackett distribution. (Yan, Singh, et al. 2018)
Copulas can be generalized to any number of random variables. A wide class of n-dimensional copula are the Archimedean copulas C(u_1,\ldots,u_n) = \phi^{-1}(\phi(u_1) + \cdots + \phi(u_n)) where \phi\colon [0,1]\to [0, \infty) is strictly decreasing, convex, with \phi(1) = 0. These also have a simple simulation algorithm.