Why the 1-Wasserstein distance is the area between the two marginal CDFs

Understand

We elucidate why the 1-Wasserstein distance $W_1$ coincides with the area between the two marginal cumulative distribution functions (CDFs).

• We first describe the Wasserstein distance in terms of copulas, and then show that $W_1$ with the Euclidean distance is attained with the $M$ copula.
• Two random variables whose dependence is given by the $M$ copula manifest perfect (positive) dependence.
• If we express the random variables in terms of their CDFs, it is intuitive to see that the distance between two such random variables coincides with the area between the two CDFs.