Suppose I pick 'N' integers over an interval [A, B] without replacement. As a function of 'N' and the interval length, what distribution / average values should I expect for the distances between nearest-neighbors in a sorted array of the selected integers?
Edit: I apologize, an important note is that the distances between the endpoints and the nearest integers to the endpoints should also be included. This is a bit like dividing a piece of rope into (B - A + 1) segments, cutting at the locations representing the 'N' selected integers, and looking at the distribution of cut rope lengths.
Edit 2: Apparently this question is in desperate need of clarification. Extending the rope example I provided, here's exactly what I'm looking for:
Upon cutting the rope into 'N' pieces, and placing these pieces in a bag, I would very much like the probability, P(k), of randomly selecting a fragment of rope of length 'k' from this bag. Here, the probability of selecting a particular fragment of the rope is independent of its length. The function for P(k) provides what I'd like to know about the distribution of rope lengths after 'N' cuts.