New Expressions for Repeated Upper Tail Integrals of the Normal Distribution

摘要

Expressions are given for repeated upper tail integrals of the univariate normal density (and so also for the general Hermite function) for both positive and negative arguments. The expressions involve moments of the form E(x + i N) n and E1 / (x 2 + N 2) n , where N is a unit normal random variable. Laplace transforms are provided for the Hermite functions and the moments. The practical use of these expressions is illustrated.