An alternative characterization of the Wasserstein metric is the following (which you do not have to show): dW (X, Y ) is the infimum of E|X − Y

|

over all possible joint distributions of (X

, Y

) such that the marginal distributions of X and Y are those of X and Y , respectively. However, do show that dW (X, Y ) =

 1 0

|F−1

(u) − G−1

(u)|du =

 ∞

−∞

|F(x) − G(x)|dx , where F and G are the c.d.f.s of X and Y , respectively.