On the Evaluation of Probabilities of Convex Polyhedra Under Multivariate Normal and t Distributions

Summary It is noted that probabilities of convex polyhedra in p-dimensional space can be expressed in terms of a function V(h1, h2, …, hp), if the distribution is normal, and in terms of a function Vn(h1, h2, …, hp), if the distribution is the t-distribution with n degrees of freedom. Formulae expressing V(h1, h2, …, hp) and Vn(h1, h2, …, hp) in terms of similar functions of lower order are obtained. An asymptotic expansion of Vn(h1,…, hp) in powers of n-1 is also included. The final section contains formulae connecting Vn(h1, h2, …, hp) with Vn±2(h1, K2, …, hp).