Closest Pair and the Post Office Problem for Stochastic Points

Given a (master) set M of n points in d-dimensional Euclidean space, consider drawing a random subset that includes each point m(i) is an element of M with an independent probability p(i). How difficult is it to compute elementary statistics about the closest pair of points in such a subset? For instance, what is the probability that the distance between the closest pair of points in the random subset is no more than l, for a given value l? Or, can we preprocess the master set M such that given a query point q, we can efficiently estimate the expected distance from q to its nearest neighbor in the random subset? We obtain hardness results and approximation algorithms for stochastic problems of this kind.