On the number of vertices of random convex polyhedra

A convex polyhedron is defined as the intersection of a finite number of closed half spaces. If the boundary hyperplanes contain random variables in the expression of their analytic definition then we have a random convex polyhedron. Thus we fix the number of half spaces but allow them to be random. The number of vertices ν of a random convex polyhedron is a random variable which we define to be equal to zero if the intersection of half spaces is empty. We are interested in the probabilistic behaviour of ν in particular to find the expectation E(ν) under various assumptions.