 # Тotal probability formula for vector gaussian distributions

https://doi.org/10.35596/1729-7648-2021-19-2-58-64

## Полный текст:

#### Аннотация

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.

### Об авторах

V. S. Mukha
Belarusian State University of Informatics and Radioelectronics
Беларусь

D.Sc., Professor, Professor at the Department of Information Technologies of Automated Systems

N. F. Kako
Belarusian State University of Informatics and Radioelectronics
Беларусь

Postgraduate student

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8. Mukha V.S., Kako N.F. Integrals and integral transformations connected with vector Gaussian distribution. Vestsі Natsyianal'nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series. 2019;55(4):457-466. DOI: org/10.29235/1561-2430-2019-55-4-457-466.

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10. Mukha V.S., Kako N.F. Dual Control of Multidimensional-matrix Stochastic Objects. Information Technologies and Systems 2019 (ITS 2019): Proceeding of the International Conference, BSUIR, Minsk, Belarus,30th October 2019. Minsk: BSUIR; 2019:236-237.

11. Mukha V.S., Kako N.F. Flat Problem of Allowance Distribution as Dual Control Problem. Information Technologies and Systems 2020 (ITS 2020): Proceeding of the International Conference, BSUIR, Minsk, Belarus, 18th November 2020. Minsk: BSUIR; 2020:195-196.

#### Для цитирования:

Mukha V.S., Kako N.F. Тotal probability formula for vector gaussian distributions. Доклады БГУИР. 2021;19(2):58-64. https://doi.org/10.35596/1729-7648-2021-19-2-58-64

#### For citation:

Mukha V.S., Kako N.F. Тotal probability formula for vector gaussian distributions. Doklady BGUIR. 2021;19(2):58-64. https://doi.org/10.35596/1729-7648-2021-19-2-58-64

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ISSN 1729-7648 (Print)
ISSN 2708-0382 (Online)