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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.

The article is devoted to the study of the seals of fixed joints. The mechanisms of leakage occurrence at different values of roughness and pressure on the joint surfaces are considered taking into account the real contact area. The estimation of deformations and forces in the contact zone was carried out taking into account the profile of surfaces, limit deviations of shape and location of surfaces, as well as physical and mechanical properties of materials of the sealed joint. The obtained theoretical dependence allows to evaluate the tightness of the fixed connection depending on the main parameters of the seal roughness, manufacturing errors and the location of the interfaces, their physical and mechanical properties and characteristics of sealed liquid, including differential pressure and the nominal pressure in the joint. A new law of probability density distribution is proposed, which more adequately describes the considered boundary conditions of the experiment. Relations for the probabilistic evaluation of the contact area and surface deformations taking into account the characteristics of the material are obtained. Simple engineering relations for the estimation of probability of contact of the projections and offset surfaces for the two profiles are obtained. The simulation of the process of leakage depending on these conditions is carried out. It is shown that the complete tightness of rigid joints is insufficient even with low roughness provided by the finishing methods of processing and in the absence of deviations in the shape of the joint surfaces. The comparison of experimental data with the obtained theoretical results showed that the proposed engineering dependences provide a good match with the experimental data.

Introduction. An advanced technique of the livestock waste disinfection is considered. The essence of the method is in the integral physicochemical action of the alternating magnetic field combined with a chemical agent. The inductor can implement this method. A residual number of the colony-forming units of total coliforms is selected as the most critical parameter characterizing the quality of the disinfection process of livestock effluents. It is proposed to use the mathematical experimental design theory to improve the efficiency of investigating the disinfection process of livestock waste. Its application implies a normal law of distribution of the value frequency of colony-forming units of total coliforms. Materials and Methods . In the course of this study, the following methods were used: direct and indirect instrumental measurement; micrographics; least quadrants, theoretical definition of the random distribution, and statistical data processing. Research Results . The hypothesis on a normal distribution law was tested by setting up an experiment at the stand with certain conditions. As a result of the conducted experiments, a sample of values consisting of 100 observations is obtained. Based on the aprior information analysis, a hypothesis on the normal distribution law of the value frequency of colony-forming units of total coliforms was adopted. Discussion and Conclusions. In the course of studying the distribution law of the value frequency of colony-forming units of total coliforms characterizing the quality of the disinfection process, the accepted critical value of the Pearson test was compared to the calculated one. Since the design value is less than the adopted critical one, the hypothesis on a normal distribution law is accepted.