Suppose we have a set of independent random variables $X_1,\ldots,X_n$ over $\mathbb{R}$.
It is easy to see that 
$$d_{ij}=E[|X_i-X_j|]$$
satisfy the triangle inequality.
Is there any study of such metric spaces?

(Note that this metric is *not* the usual metric of distributions. In particular,
for two identically distributed $X_1,X_2$, $d_{12}\ne 0$.)