  In this paper, we study the expectation values of topological invariants of
the Vietoris-Rips complex and \v{C}ech complex for a finite set of sample
points on a Riemannian manifold. We show that the Betti number and Euler
characteristic of the complexes are Lipschitz functions of the scale parameter
and that there is an interval such that the Betti curve converges to the Betti
number of the underlying manifold.
