For a simple, finite, connected and complete graph $K_n = (V(K_n), E(K_n))$ with vertex set $V(K_n)$ and edge set $E(K_n)$, we assign a non-negative independent and identical distributed random weight $\xi_e$ with distribution $F$ to each edge $e \in E(K_n)$. We also assign a probability of appearance $0\le P_v \le 1$ to each $v \in V(K_n)$ as well. Then what is the expected lengths of the minimum spanning trees?

In case of no probability on the vertices this problem was solved by Wenbo Li and Xinyi Zhang in 2008.