  We consider the random hypergraph on a finite vertex set by choosing each set
of vertices as an hyperedge independently at random. We express the probability
distributions of the (lower-)associated simplicial complex and the
(lower-)associated independence hypergraph of the random hypergraph in terms of
the probability distributions of certain random simplicial complex and certain
random independence hypergraph of Erd\"os-R\'enyi type. We construct a graded
structure of the map algebra explicitly and give algorithms to generate random
hypergraphs and random simplicial complexes.
