  Assume that a finite set of points is randomly sampled from a subspace of a
metric space. Recent advances in computational topology have provided several
approaches to recovering the geometric and topological properties of the
underlying space. In this paper we take a statistical approach to this problem.
We assume that the data is randomly sampled from an unknown probability
distribution. We define two filtered complexes with which we can calculate the
persistent homology of a probability distribution. Using statistical estimators
for samples from certain families of distributions, we show that we can recover
the persistent homology of the underlying distribution.
