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

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

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 $X_1,X_2$, $d_{12}\ne 0$.)