A generalization for the expected value of the earth mover’s distance

The earth mover's distance (EMD), also called the first Wasserstein distance, can be naturally extended to compare arbitrarily many probability distributions, rather than only two, on set $[n]=\{1,\dots,n\}$. We present details for this generalization, along with a highly efficient algorithm inspired by combinatorics; it turns out that in special case of three EMD is half sum pairwise EMD's. Extending methods Bourn and Willenbring (arXiv:1903.03673), we compute expected value generalized random $d$-tuples using generating function which coincides Hilbert series Segre embedding. then use analyze real-world data grade distributions.