Multivariate Gaussian distribution is one of the most important continuous distributions. If some components are restricted to an interval, either finite or semi-finite, it is referred to as the truncated multivariate normal (TMVN) distribution. Many statistical algorithms rely on the evaluation of some expectations with respect to a TMVN, especially in the expectation-maximization (EM) type algorithms. In this talk, we present a fast hierarchical algorithm which can reduce the computational complexity of evaluating a class of p-dimensional TMVN probability and expectation integrals to the asymptotically optimal O(p), by utilizing the low-rank and low-dimensional structures in the inverse of the covariance matrix, and by processing the compressed information efficiently on a hierarchical tree structure.