Abstract
In many applications it is desirable to consider not one random vector but a number of random vectors with the joint distribution. This paper is devoted to the integral and integral transformations connected with the joint vector Gaussian probability density function. Such integral and transformations arise in the statistical decision theory, particularly, in the dual control theory based on the statistical decision theory. One of the results represented in the paper is the integral of the joint Gaussian probability density function. The other results are the total probability formula and Bayes formula formulated in terms of the joint vector Gaussian probability density function. As an example the Bayesian estimations of the coefficients of the multiple regression function are obtained. The proposed integrals can be used as table integrals in various fields of research.
Highlights
The integrals connected with probability distributions are used in many applications, one of them being the statistical decision theory, or, in other words, the Bayesian approach in statistics [1,2,3,4]
The equations of dynamic programming in the dual control problem contain the integrals connected with the multivariate probability distributions
I =1, m, the posterior probability density function f(ξ/y) of the random vector Ξ defined by the Bayes formula (28), has the following form:
Summary
The integrals connected with probability distributions are used in many applications, one of them being the statistical decision theory, or, in other words, the Bayesian approach in statistics [1,2,3,4]. The statistical decision theory attracts much attention due to the ability to formulate problems in strict mathematical form and to process data in real time. The equations of dynamic programming in the dual control problem contain the integrals connected with the multivariate probability distributions. The integrals and integral transformations connected with the vector Gaussian distribution were considered in papers [6, 7]. Integrals connected with the joint vector Gaussian distribution. The parameters vΞ and dΞ are the mathematical expectation and variance-covariance matrix of the random vector Ξ respectively
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