Probability and Statistical Theory for Applied Researchers

The Correspondence Between A. A. Markov and A. A. Chuprov.- A Review of the Correspondence Between A. A. Markov and A. A. Chuprov.- Appendices: A. A. Markov and A. A. Chuprov on the Law of Large Numbers.- Appendix 1. On the Basic Principles of the Calculus of Probability and on the Law of Large Numbers A. A. Markov.- Appendix 3. The Bicentennial of the Law of Large Numbers A. A. Markov.- Appendix 4. The Law of Large Numbers in Contemporary Science A. A. Chuprov.

Controlled Branching Processes

According to the current situation of the local colleges and the theory of probability and mathematical statistics in teaching practice, how to design teaching content and teaching methods, etc, the paper puts forward the suggestions and measures in teaching reform for this course.

The experience of usage of project method in teaching such disciplines as “Mathematics”, “Theory of Probability and Mathematical Statistics” for economic courses of higher education (specialist`s degree and bachelor degree) has been described the article. There is a brief analysis of the origin and development of project method, its modern classifications. Since these subjects are studied during the first and second year, the main difficulties in the implementation of project method are low research skills and teamwork of learners. Nevertheless, the authors suggest it is a group project, which has a number of advantages over the individual work. For example, the topic "Statistical estimation of the distribution parameters" presents the author´s approach to the implementation of the project activities. The structure of the project, the main stages of its formation and implementation are considered. The conclusion about the effectiveness of project method is done by comparing the control and experimental groups.

The main problem of special aerial reconnaissance is the contradiction between the dimensions of the survey area and the dimensions of elementary objects (targets). Since the viewing area is very large, the size of the target is small, and the instantaneous field of view of the on-board equipment is limited, the time for obtaining data is quite significant and the amount of information to be processed is very large. Also, due to technical limitations on the time spent in the air and the limitations of the reconnaissance equipment, one UAV is not able to cover the entire area in one flight. The severity of this problem can be partially removed by using a group of UAVs. but both for one person and for a group, the task of planning the route of movement arises. A simple comb survey of the reconnaissance zone is far from optimal, and in practice route planning is used by points with the involvement of a priori information about the most likely locations of targets. However, this method is rather inconvenient for the automatic route planning of the UAV itself, since the point on the map usually does not correlate with the flight height, the instantaneous field of view of the on-board surveillance equipment, the scale and detail of the picture. In order to increase the optimization possibilities of automatic flight planning, it is suggested to use the target distribution density function (TDF). The density function of the target distribution is a two-dimensional mathematical model that describes the conditional relative probability of finding targets at different points in space. It is created and set on the basis of a priori data about the terrain, previous observations or intellectual assessments, reflecting the distribution of possible targets in a certain area or for the entire reconnaissance area. TDF allows to model the space not as homogeneous, but as an area with different degrees of importance or probabilities of finding targets. It should be noted that obtaining mathematically justified or calculated by the theory of probability or mathematical statistics, two-dimensional discrete probability densities for randomly placed targets is actually impossible. Therefore, here the TDF is considered as an area with different degrees of review importance, which is reflected by conditional relative probabilities obtained from various sources, including expert evaluation.

Summary of Reports Presented at Sessions of the Seminar on the Theory of Probability and Mathematical Statistics at the Mathematical Institute of the Academy of Sciences of the USSR, 1978

Summary of Reports Presented at Sessions of the Seminar in the Theory of Probability and Mathematical Statistics at the Kiev Shevchenko State University 1981

The paper presents an application of the Automated Independent Replication with Gathering Statistics of the Stochastic Processes Method in achieving and controlling the accuracy of simulation results in the Monte Carlo queuing simulations. The method is based on the application of the basic theorems of the theory of probability and mathematical statistics. The accuracy of the simulation results is linked with a number of independent replications of simulation experiments.

Егоров, О. И. Усовершенствование процедуры определения количества и осности подвижных единиц / О. И. Егоров // Транспортні системи та технології перевезень : зб. наук. пр. Дніпропетр. нац. ун-ту залізн. трансп. ім. акад. Лазаряна. — Дніпропетровськ, 2015. — № 10. — С. 38—43. — doi: 10.15802/tstt2015/57064.

Most of the objects to be found in nature are essentially random, i.e. affecting the probing optical radiation they transform its amplitude and phase parameters in complex, hardly predicted manner. Traditionally, such objects are described in the framework of the theory of probability and mathematical statistics. In last decade, however, the novel approach is outlined according to which rather extensive class of natural and artificial objects can be represented as any mixture of deterministic and random spatial structures. The fractals, i.e. the objects possessing the feature of structural self-similarity in the strict or in any generalized sense, are the most adequate model describing such objects'. Interaction of these objects with a coherent radiation results in stochastic spatial structures. Here we consider some techniques Lor structural diagnostics of an object that are based both on the probabiitic approach 'and on the stochastic one. Speaking about random optical fields, we will imply spatially inhomogeneous stationary fields resulted from interaction of coherent radiation with random phase or amplitude objects. One encounters such a speckle field solving the problem of remote structural diagnostics of an object. Besides, we will limit our consideration with the scalar approximation. Surface roughness is often considered as the classic model of the object causing spatial modulation of the probing beam. On the one hand, it is easily to prepare such an object with the independently controlled parameters. On the other hand, the diagnostical problem is of the most practical importance for the objects of this kind. That is why, we will illustrate the capabilities of the optical correlometric techniques using a rough surface as the basic model, as a rule. At the same time, amplitude and phase deterministic or quasi-deterministic objects will be also under our consideration, and the fractals whose main feature consists in their structural self-similarity' will be the most significant examples of these objects. It is well known, that genera! solution of a statistical diagnostica! problem consists in reconstruction of the height distribution function for a rough surface inhomogeneities or, alternatively, in finding out the complete set of the statistical moments characterizing the object's structure. If the object is assumed to be random in the strict sense, then its description in terms of a height distribution function or in terms of a complete set of the statiical moments represents comprehensive solution of the rtatistica! diagnostical problem. In practice, however, necessity in such a general statistical description occurs to be rather rare. In accordance with the well known statements of mathematical statistics, knowing of several low-order statistical moments is enough for characterization of the object under study in its significant features and in desirable approximation. In the case of rough surfaces, the correlation length and root-mean-square (rms) of roughness from the mean surface line (that is linked by simple relation with a phase variance of the boundary field in some practically important cases) are the main statistical parameters adopted for rough surface characterization. As any simple model for surface roughness is used, particularly, by presuming Gaussian or another symmetrical height distribution function, the diagnostical problem is solved in principle by determining these parameters through any of the known high efficient techniques2. For all that, the question arises: if the experimentally found parameters are corresponding with desirable accuracy to the intrinsic roughness parameters? In other words, the problem of adequacy of the used model for a rough surface arises. It is stated in some recently reported studies that the fractal model occurs to be adequate for rough surface characterization in numerous practical applications23. So, particularly, for a machined surface finish such a conclusion is grounded in Ref.2 . It is not less important, attraction of the approaches of a chaos theory as well as fractal optics methodology enables to extend the circle of the problems of rough surface diagnostics that can be solved in the framework of the unified approach, irrespective of the phase variance at the boundary field. Of course, in the framework of this approach, the solution of a diagnostical problem is formulated in unconventional terms, such as chaos dimension closely related to the notion of entropy. The diagnostically important interconnection among the dimension parameters of the scattered optical radiation and the corresponding dimension parameters of the object's structure is found. Somewhat (to say, in the mentioned above surface finish problem) unambiguous interconnections occur to be put in evidence among these (dimensional) parameters with the traditional ones, such as the standard deviation of the surface profile from the mean surface line and the correlation length of surface inhomogeneities2. This paper is devoted to consideration of the present state of the problem briefly outlined above and interrelations among different mentioned approaches.

The article deals with the problem of introducing cloud calculations into 10th-11th graders’ training to solve optimization problems in the context of the STEM-education concept. After analyzing existing programmes of optional courses on optimization problems, the programme of the optional course Optimization Problems has been developed and substantiated implying solution of problems by the cloud environment CoCalc. It is a routine calculating operation and not a mathematical model that is accentuated in the programme. It allows considering more problems which are close to reality without adapting the material while training 10th-11th graders. Besides, the mathematical apparatus of the course which is partially known to students as the knowledge acquired from such mathematics sections as the theory of probability, mathematical statistics, mathematical analysis and linear algebra is enough to master the suggested course. The developed course deals with a whole class of problems of conventional optimization which vary greatly. They can be associated with designing devices and technological processes, distributing limited resources and planning business functioning as well as with everyday problems of people. Devices, processes and situations to which a model of optimization problem is applied are called optimization problems. Optimization methods enable optimal solutions for mathematical models. The developed course is noted for building mathematical models and defining a method to be applied to finding an efficient solution.

Providing growing economic needs is impossible without economic development. Economic development is associated with qualitative and structural changes in the economy, education, science, and culture. In accordance with the changes in society, the purpose of economic science also changes. There is a transition from observing the economy to influencing economic processes, their management, planning, forecasting, implementation of specific tasks at individual enterprises within certain terms in accordance with the general goals and interests of the national economy. The level of influence on economic phenomena varies from recommendations to specific methods of quantitative calculation, which provide an objective choice of an economic decision. Scientific management of economic processes can be carried out only on the basis of the application of precise mathematical methods in all spheres of the economy – from the distribution of minerals to the demand for consumer goods and household services, labor needs, planning of logistical tasks, etc. Therefore, the subject of the research is conducting mathematical modeling of an economic phenomenon, analysis of economic processes, using the methodology of economic-mathematical modeling and mathematical programming, with the aim of forecasting and studying one or another aspect of its development in the planning and management system, in information provision, in the possibility taking into account the dynamics of economic processes. The study of various economic phenomena with the help of economic-mathematical models, which are a simplified description and take into account the most significant and determining factors of the studied phenomenon, involves the use of mathematics for a formal description of the most significant relationships between economic variables and objects: conclusions can be obtained by the method of deduction, which are adequate for the object under study; methods of mathematics and statistics make it possible to obtain new knowledge about the researched object; the use of mathematical terminology allows you to accurately and compactly express the statements of economic theory, formulate its concepts and conclusions. The analysis of previous studies shows that the problems of organizing the teaching of mathematical disciplines in higher educational institutions, taking into account modern requirements, were taken care of by scientists: I.P. Vasylchenko, L.P. Husak, G.Ya. Dutka, T.V. Krylova, O.V. Levchuk, L.I. Nichugovska, V.A. Petruk et al. The need to build a holistic methodology for the introduction of higher mathematics, the theory of probabilities and mathematical statistics, mathematical modeling and optimization of economic processes is the key task of the article on the use of economic and mathematical models for the development and implementation of optimal projects for the development of the production and services market. Conclusion: to solve effectively a wide class of problems with appropriate solution methods, covering various problems of development and functioning of real economic systems, it is convenient to develop economic and mathematical models, which, in combination with powerful, high-speed computing equipment and modern software products, will make up decision-making support systems in various sectors of the economy.

The theory of statistics, like other scientific theories, can be divided into three fairly distinct sections; they are (1) the foundations of the theory, (2) the superstructure, and (3) the principles of application. It is, of course, possible to divide the subject in different ways; for example, the word ‘methodology’ is sometimes used to cover part of (2) and (3). But I find it helpful in thinking about the subject to make the division that I have indicated.It is my purpose to show that actuarial science is essentially a particular application of the theory of statistics, that there is nothing very special about its superstructure and that in its application to practical affairs the universal principles apply. But I first want to make it clear that while I include the combination of compound interest functions with other actuarial functions as part of actuarial science I do not regard finance and investment as part of actuarial science proper. While treating finance and investment as a part of economics I appreciate that they represent an essential ingredient in the equipment of the practising actuary.

Changes in the natural gas market related to the implementation of the requirements of the European Union's Third Energy Package affect the financial security of gas distribution companies. Changes in the natural gas market related to the implementation of the requirements of the European Union's Third Energy Package affect the financial security of gas distribution companies. The purpose of the article is to develop a scientific and methodological approach to the selection of an organizational and economic mechanism for ensuring the financial security of gas distribution enterprises, which takes into account the combination of their status as natural monopoly entities and enterprises, and to substantiate practical recommendations for its implementation. A set of methods of scientific cognition was used, namely: the fundamental provisions of the theory of finance, probability theory and mathematical statistics, and the method of adaptive forecasting. The choice of the optimal type of organizational and economic mechanism for ensuring the financial security of gas distribution enterprises is proposed to be carried out taking into account the phased withdrawal from the financial crisis. If the type of mechanism is used effectively, the enterprise proceeds to implement a preventive mechanism; if it is ineffective, it is necessary to review the measures or proceed to the liquidation procedure; if the preventive mechanism is used effectively, the enterprise proceeds to the stabilization mechanism or a radical one. The scientific and methodological approach to the choice of an organizational and economic mechanism for ensuring the financial security of gas distribution enterprises in Ukraine has been proposed. It is aimed at consistent and targeted implementation of a set of preventive, stabilization and radical measures that enables prevention of financial risk, timely assessment and forecasting of its level, as well as development, implementation and monitoring of the relevant program. The implementation of the proposed scientific and methodological approach in the practical activities of gas distribution enterprises involves the preparation of substantive information, analytical and methodological support, which consists in a comprehensive analysis of financial security.

Identification became a quite popular topic in Information Theory. It is a very good example of interaction between Mathematical Statistics (Testing of Hypotheses) and Information Theory.