Тotal probability formula for vector Gaussian distributions
The total probability formula for continuous random variables is the integral of product of two probability density functions that defines the unconditional probability density function from the conditional one. The need for calculation of such integrals arises in many applications, for instant, in...
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| Format: | Article |
| Language: | Russian |
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Educational institution «Belarusian State University of Informatics and Radioelectronics»
2021-03-01
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| Series: | Doklady Belorusskogo gosudarstvennogo universiteta informatiki i radioèlektroniki |
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| Online Access: | https://doklady.bsuir.by/jour/article/view/3035 |
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| _version_ | 1849240183876091904 |
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| author | V. S. Mukha N. F. Kako |
| author_facet | V. S. Mukha N. F. Kako |
| author_sort | V. S. Mukha |
| collection | DOAJ |
| description | The total probability formula for continuous random variables is the integral of product of two probability density functions that defines the unconditional probability density function from the conditional one. The need for calculation of such integrals arises in many applications, for instant, in statistical decision theory. The statistical decision theory attracts attention due to the ability to formulate the problems in a strict mathematical form. One of the technical problems solved by the statistical decision theory is the problem of dual control that requires calculation of integrals connected with the multivariate probability distributions. The necessary integrals are not available in the literature. One theorem on the total probability formula for vector Gaussian distributions was published by the authors earlier. In this paper we repeat this theorem and prove a new theorem that uses more familiar form of the initial data and has more familiar form of the result. The new form of the theorem allows us to obtain the unconditional mathematical expectation and the unconditional variance-covariance matrix very simply. We also confirm the new theorem by direct calculation for the case of the simple linear regression. |
| format | Article |
| id | doaj-art-c8b36b90ad3f4a5c9f8c3cc10aabe776 |
| institution | Kabale University |
| issn | 1729-7648 |
| language | Russian |
| publishDate | 2021-03-01 |
| publisher | Educational institution «Belarusian State University of Informatics and Radioelectronics» |
| record_format | Article |
| series | Doklady Belorusskogo gosudarstvennogo universiteta informatiki i radioèlektroniki |
| spelling | doaj-art-c8b36b90ad3f4a5c9f8c3cc10aabe7762025-08-20T04:00:41ZrusEducational institution «Belarusian State University of Informatics and Radioelectronics»Doklady Belorusskogo gosudarstvennogo universiteta informatiki i radioèlektroniki1729-76482021-03-01192586410.35596/1729-7648-2021-19-2-58-641681Тotal probability formula for vector Gaussian distributionsV. S. Mukha0N. F. Kako1Belarusian State University of Informatics and RadioelectronicsBelarusian State University of Informatics and RadioelectronicsThe total probability formula for continuous random variables is the integral of product of two probability density functions that defines the unconditional probability density function from the conditional one. The need for calculation of such integrals arises in many applications, for instant, in statistical decision theory. The statistical decision theory attracts attention due to the ability to formulate the problems in a strict mathematical form. One of the technical problems solved by the statistical decision theory is the problem of dual control that requires calculation of integrals connected with the multivariate probability distributions. The necessary integrals are not available in the literature. One theorem on the total probability formula for vector Gaussian distributions was published by the authors earlier. In this paper we repeat this theorem and prove a new theorem that uses more familiar form of the initial data and has more familiar form of the result. The new form of the theorem allows us to obtain the unconditional mathematical expectation and the unconditional variance-covariance matrix very simply. We also confirm the new theorem by direct calculation for the case of the simple linear regression.https://doklady.bsuir.by/jour/article/view/3035total probability formulavector gaussian distributionmultivariate integralsmultivariate regression |
| spellingShingle | V. S. Mukha N. F. Kako Тotal probability formula for vector Gaussian distributions Doklady Belorusskogo gosudarstvennogo universiteta informatiki i radioèlektroniki total probability formula vector gaussian distribution multivariate integrals multivariate regression |
| title | Тotal probability formula for vector Gaussian distributions |
| title_full | Тotal probability formula for vector Gaussian distributions |
| title_fullStr | Тotal probability formula for vector Gaussian distributions |
| title_full_unstemmed | Тotal probability formula for vector Gaussian distributions |
| title_short | Тotal probability formula for vector Gaussian distributions |
| title_sort | тotal probability formula for vector gaussian distributions |
| topic | total probability formula vector gaussian distribution multivariate integrals multivariate regression |
| url | https://doklady.bsuir.by/jour/article/view/3035 |
| work_keys_str_mv | AT vsmukha totalprobabilityformulaforvectorgaussiandistributions AT nfkako totalprobabilityformulaforvectorgaussiandistributions |