Integrals and integral transformations connected with vector Gaussian distribution

Abstract

The paper is devoted to the integrals and integral transformations related to the multivariate Gaussian probability density function. Such integrals and integral transformations arise in probability applications. In the paper, three integrals are presented which allow calculation the moments of the multivariate Gaussian distribution. Besides, the total probability formula and Bayes formula for vector Gaussian distributions are given. The proofs of the obtained results are given. The proof of the integrals is performed on the base of Gauss elimination method. The total probability formula and Bayes formula are obtained on the base of the proved integrals. These integrals and integral transformations could be used, for instant, in the statistical decision theory, particularly, in the dual control theory, and as the table integrals in various areas of research. On the basis of the obtained results, Bayesian estimations of the coefficients of the multiple regression function are calculated.