We present a novel analytical approach to computing the population and geometric parameters of a multi-robot system that will provably produce specified boundary coverage statistics. We consider scenarios in which robots with no global position information, communication, or prior environmental data have arrived at uniformly random locations along a simple closed or open boundary. This type of scenario can arise in a variety of multi-robot tasks, including surveillance, collective transport, disaster response, and therapeutic and imaging applications in nanomedicine. We derive the probability that a given point robot configuration is saturated, meaning that all pairs of adjacent robots are no farther apart than a specified distance. This derivation relies on a geometric interpretation of the saturation probability and an application of the Inclusion-Exclusion Principle, and it is easily extended to finite-sized robots. In the process, we obtain formulas for (a) an integral that is in general computationally expensive to compute directly, and (b) the volume of the intersection of a regular simplex with a hypercube. In addition, we use results from order statistics to compute the probability distributions of the robot positions along the boundary and the distances between adjacent robots. We validate our derivations of these probability distributions and the saturation probability using Monte Carlo simulations of scenarios with both point robots and finite-sized robots.