Probability Theory Research Articles

INCREASE OF TECHNOLOGICAL INDICATORS USED IN THE ENERGY INDUSTRY

When evaluating the suitability of using coal in various industries, it is subjected to technical analysis and a number of special technological studies. The obtained results in combination with elemental and group analysis allow choosing the most rational and economically efficient directions of their use, which plays not the least role in determining the economic significance of coal mining deposits. The work is based on theoretical research methods based on the basics of probability theory and mathematical statistics - to improve the existing methods of processing results and assessing the uncertainty of measurements; theories of accuracy - for the development of procedures for assessing the uncertainty of measurements of the quality parameters of solid mineral fuel during its tests; statistical modeling methods - to check the reliability of the developed provisions. A generalized parameter of the quality of solid mineral fuel has been synthesized, which makes it possible to evaluate its quality based on the results of tests conducted in coal chemical laboratories. improvement of methods of testing solid mineral fuel according to quality parameters by developing uncertainty assessment procedures.

Computing the Noncommutative Inner Rank by Means of Operator-Valued Free Probability Theory

AbstractWe address the noncommutative version of the Edmonds’ problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued quadratic equation, for which we provide precise analytical and numerical control on the fixed point algorithm for solving the equation. Numerical examples show the efficiency of the algorithm.

Beyond risk preferences in sequential decision-making: How probability representation, sequential structure and choice perseverance bias optimal search

Sequential decision-making, where choices are made one after the other, is an important aspect of our daily lives. For example, when searching for a job, an apartment, or deciding when to buy or sell a stock, people often have to make decisions without knowing what future opportunities might arise. These situations, which are known as optimal stopping problems, involve a risk associated with the decision to either stop or continue searching. However, previous research has not consistently found a clear connection between individuals’ search behavior in these tasks and their risk preferences as measured in controlled experimental settings. In this paper, we explore how particular characteristics of optimal stopping tasks affect people’s choices, extending beyond their stable risk preferences. We find that (1) the way the underlying sampling distribution is presented (whether it is based on experience or description), (2) the sequential presentation of options, and (3) the unequal frequencies of choices to reject versus to accept significantly bias people choices. These results shed light on the complex nature of decisions that unfold sequentially and emphasize the importance of incorporating context factors when studying human decision behavior.

Justification Models of Organizational and Technical Support of Measures to Create Information Protection System of Informatization Objects

The relevance of the article is due to the need to protect information at all stages of creating an information system in the absence of appropriate organizational and technical support. To justify this kind of provision, mathematical models are needed that take into account the conditions and the random nature of the implementation of heterogeneous information processing processes in time at each stage, threats to its security and protection from these threats. The article discusses approaches to the development of such models based on the apparatus of flow theory and the theory of composite Petri ‒Markov networks, the possibilities of which for modeling the processes under study have not been previously considered. The purpose of the article. To reveal the content of issues related to the justification of organizational and technical support for information protection at the stages of creating an information system, to show the need and ways to quantify threats to its security and protection from these threats. For this purpose, the methods of functional and structural analysis, probability theory and flow theory were applied. In the course of solving the scientific problem, the relevance of substantiating the organizational and technical support of information protection at the stages of creating information systems is shown, descriptive models of its processing processes are developed, time diagrams of threat scenarios are given in the absence of protection measures and in the conditions of using preventive organizational and technical measures, indicators and analytical ratios for their calculation are proposed. The scientific novelty of the article consists in the fact that for the first time it examines the issues of organizational and technical support for information protection at the stages of creating an information system, examines the theoretical aspects of quantifying threats to its security and protection from these threats based on flow theory and prospects for applying the theory of composite Petri-Markov networks. Significance (theoretical). The conditions for the implementation of the studied processes and the possibility of applying the theory of flows and the apparatus of composite Petri–Markov networks for their modeling are established. Significance (practical). The results obtained in the work can be used to justify the organizational and technical provision of information protection at the stages of creating information systems of various organizations (both state and non-state), for which the implementation of threats at these stages may lead to damage to their activities.

Some types of problems from probability theory

Some types of problems from probability theory

International organizations’ practice in the interpretation of their constituent instruments

Abstract Subsequent practice of international organizations plays an important role in the interpretation of their constituent instruments. Based on Articles 5 and 31(3) of the Vienna Convention on the Law of Treaties (VCLT), when such practices have become established or have been consented to by all States parties, they can be considered authoritative interpretations. The legal basis for considering practices of international organizations in other circumstances, however, is not entirely clear. The International Law Commission proposed that these practices may still be considered in the application of Articles 31 and 32 of the VCLT but left many issues undiscussed. This article argues that it is possible to consider international organizations’ practice in the application of Article 31 if a strong inference about the original intention of States parties can be drawn therefrom. To address difficulties involved in evaluating the probative value of international organizations’ practice, this article borrows a tool from probability theory—likelihood ratio—and shows that, absent a major change of circumstances, international organizations’ earlier practice can be seen as credible evidence of States parties’ original intention and considered in the application of Article 31.

Tail probability bounds for acceptable random variables of partial sums and their application in linear models

Exponential probability inequalities play an important role in many parts of probability and statistics, offering a fundamental tool for tail probability bounding and modeling stochastic processes. The present work proposes a new exponential inequality for acceptable random variables. This class covers dependencies that are not embraced by the standard independence assumptions. Our results extend the theoretical perspective by placing sharper bounds on the above-mentioned variables, which tends to increase the accuracy of probabilistic modeling. Moreover, we study complete convergence in first-order autoregressive processes with acceptable innovations. The complete convergence framework gives further insights into the long-term behavior of the wide class of processes commonly used in time series analysis, econometrics, and other areas of applied research. In turn, our results on the influence of acceptable innovations on the properties of complete convergence establish bridges between theoretical probability inequalities and statistical applications. In addition, these contributions provide new tools in investigating the stability and predictability of the autoregressive models. These considerations consequently set up significant perspectives and enrich a developing body of research related to stochastic processes by reinforcing the theoretical framework as well as the methodological apparatus of probability theory.

NEUTROSOPHIC EXTENSION OF THE NEGATIVE BINOMIAL DISTRIBUTION: PROPERTIES AND APPLICATIONS TO INTERVAL-VALUED DATA

In real world while addressing the challenges of ambiguity and vagueness, the problems call for a scientific approach to quantify the uncertainties. The negative binomial distribution, a cornerstone in classical probability theory, traditionally examines the probability of achieving the kth success in n trials, following failures, with precise observations. However, the available literature does not address the occurrence of observations in the intervals and hence the handling of interval-valued data remained unattended. This study introduces the neutrosophic negative binomial distribution which is extended from the classical version by incorporating both indeterminacy and crisp intervals. We derive various properties of this generalized distribution which includes; characteristic function, probability function and moment generating function. This study also presents the reliability analysis metrics like cumulative hazard rate, Mills ratio, survival hazard rate, odds ratio and reversed hazard rate. This study explores the order statistics for neutrosophic negative binomial distribution, encompassing with joint, maximum, minimum and median order statistics. Special case studies are discussed which demonstrate the application of the proposed distribution, paving the way for addressing complex problems that combine conventional approaches with non-precisely determined information.

Carbon option pricing and carbon management under uncertain finance theory

Against the backdrop of increasingly severe global climate change issues, more and more countries and regions are adopting carbon emission trading, which is an effective market mechanism. As an important component of the carbon market, American carbon options provide more flexible trading tools to help companies avoid price risks. Therefore, the pricing research of American carbon options is of great significance for promoting the healthy development of the carbon market. Nonetheless, existing pricing methods are all based on probability theory, which need to satisfy the premise that the distribution function is close to the true frequency sufficiently. However, usually such premise is ill-suited in carbon market. To alleviate the predicament of the need for satisfying probability theory’s premise, this work investigates American carbon option pricing under the framework of uncertainty theory, which is another axiomatic mathematical system. For this purpose, uncertain differential equation is adopted to model the price of carbon future, and pricing formulae for American call and put carbon options are derived based on this model. Real data analyses illustrate the proposed method in details. Furthermore, it provides a detailed schema of applying the American carbon option for carbon asset management, which can optimize risk control for carbon consuming enterprises.

Novel family of probability generating distributions: Properties and data analysis

Abstract Current probability distributions often fall short when modeling lifespan data with non-monotonic hazard rate shapes, highlighting a gap in probability theory. To address this issue, this study introduces the novel family of probability-generating distributions (NP-G) as a solution to capture complex data patterns better. This research aims to develop and validate the NP-G distributions, which include a two-parameter continuous model based on the Weibull distribution. The study extensively explores the mathematical properties of the NP-G distributions, including series expansions of probability density functions, and evaluates various probabilistic measures such as moments, stress strength, mean deviations, entropy measures, and order statistics. Using the maximum likelihood method for parameter estimation and conducting a numerical simulation analysis, the study assesses the performance of the proposed extended exponentiated Weibull model. The findings demonstrate that this new model offers superior fit and accuracy compared to existing distributions when applied to two referenced datasets. This advancement is significant as it provides a more effective tool for analyzing lifespan data with non-monotonic hazard rates, enhancing both theoretical understanding and practical applications in fields such as reliability engineering and survival analysis.

An Overview of the Basic Theory of Probability Theory and its Commercial Application

Abstract. Probability theory is a key field of mathematics that focus on developing theoretical models and mathematical frameworks to describe and analyze random events or processes. As a part of basic science, probability theory holds a central place in pure mathematics while also serving a crucial function across various domains, including natural science, social science and engineering technology. This paper summarizes the main concepts, basic theories and commercial applications of probability theory. Probability theory encompasses ideas such as random experiment, sample space, event, probability, conditional probability, independence, Bayes Theorem, random variable, probability distribution, expected value and variance, law of large numbers and central limit theorem. This paper provides an overview of key concepts and theorems from probability theory, along with their practical applications in everyday life, in order to make people deeply recognize the broad and profound contents and research fields of probability theory and its wide application in various fields as an academic tool, which has a great relation with people.

Realization of an Arbitrary Structure of Perfect Distinguishability of States in General Probability Theory

Realization of an Arbitrary Structure of Perfect Distinguishability of States in General Probability Theory

Development of mathematical models of power consumption at coal plants

The object of research is coal enterprises where powerful electrical equipment is used (underground installations up to 3000 kW, tunneling complexes of 500–1500 kW, technological complexes of 6–10/0.4 kV, etc.). The deficit of generated capacities, caused by growing energy consumption, can be reduced by regulating power consumption modes. The relevance of this issue is determined by the need to conduct a research of the electricity consumption system, to determine the qualitative and quantitative characteristics of electricity consumption using mathematical models. The lack of mathematical models makes it difficult to analyze energy intensity and consumption modes of each technological operation in the overall balance of electricity consumption of coal enterprises. The article considers the structure and classification of the main technological groups of energy consumers, the development of mathematical models for each type of load modes, as well as a generalized model of electricity consumption of coal enterprises, with the use of mathematical apparatus of probability theory and mathematical statistics. As a result of the work, mathematical models of the electricity consumption process for the main technological groups and for models of daily electricity consumption of coal enterprises as a whole were developed, and it was also established that technological objects of electricity consumption are divided into three different values, in terms of power consumption modes: constant, uniform and pulse. For each class of consumers their statistical characteristics were obtained. The work results can be applied for managing power consumption modes of coal enterprises

Effective use of the tools of probability theory and mathematical statistics in solving the problems of the professional orientation of physicists and mathematicians: an interdisciplinary approach

ABSTRACT The use of methods of mathematical statistics for teaching students carries the task of determining mathematical abilities, formalizing the choice of a professional orientation and the potential choice of solutions. In higher education, the use of the mathematical apparatus allows students to form the ability to multiple choice and assess the conditions of the choice. These include not only the choice of training courses, but also the areas of knowledge in which each student can fulfil themself. For mathematicians, this is especially true, since in most cases mathematics is used to solve purely applied problems. The novelty of the study is determined by the fact that the methods and tools of probability theory cannot be applied exclusively on an individual level. To determine the effectiveness of the methods used, it is necessary to determine the possibilities of group application of the used methodological apparatus and identify areas of assistance to the student in choosing a professional orientation. In terms of the use of ubiquitous distance education, the authors demonstrate the possibility of group work with mathematical support through the use of clusters and network techniques, when each individual is the host of the training programme.

Six lectures on linearized neural networks

Abstract This tutorial examines what can be learnt about the behavior of multi-layer neural networks from the analysis of linear models. While there are important gaps between neural networks and their linear counterparts, many useful lessons can be learnt by studying the latter. A few preliminary remarks, before diving into the math: • We will not assume specific background in machine learning, let alone neural networks. On the other hand, we will assume some graduate-level mathematics, in particular probability theory (however, we will refer to the literature for complete proofs.) • Some of the notations that are used throughout the text will be summarized in appendix A. • We will keep bibliographic references in the main text to a minimum. A short guide to the literature is given in appendix B.

Computation and Applications of Gaussian Integrals in Mathematics and Applied Sciences

Gaussian integrals, particularly those involving the function e−x2 , play a central role in various fields, ranging from physics to finance. This paper explores the computation of Gaussian integrals, beginning with the fundamental integral in the interval −∞ to ∞, which yields π . The author extends this analysis to integrals involving powers of x multiplied by e−x2 , showing their relevance in calculating moments in quantum mechanics and probability theory. Additionally, the author discusses the applications of multivariate Gaussian integrals in machine learning and statistical mechanics, where they are key to solving problems in high-dimensional spaces. Practical examples are provided from quantum mechanics (path integrals), statistical mechanics (partition functions), finance (option pricing models), and machine learning (Gaussian processes). Through these examples, the paper highlights the versatility and universality of Gaussian integrals as essential tools in both theoretical and applied contexts. The integral’s widespread applicability reflects its importance in connecting mathematical theory with real-world phenomena. This work highlights the role played by the Gaussian integral.

Probability Updating in the Classical and the Quantum Model

Imagine a Bayesian decision agent who is keen to invest only in technologies or businesses that are conductive to achieving a sustainable economic future. Being initially keen to invest in a certain technology, subsequently two pieces of new information are received that both individually reduce the agent’s inclination to make that investment. In such a situation, it would be natural to assume that the simultaneous consideration of both pieces of information should further reduce the agent’s initial enthusiasm; after all the Bayesian method has been called “ nothing but common sense reduced to calculation.” Somewhat surprisingly, making general statements like this about the double conditional probability requires substantial additional assumptions in classical Bayesian probability theory. We investigate four schemes of assumptions that allow ‘reasonable’ conclusions about the double conditional probability. We compare this with two schemes for the quantum probability model where the double conditional results from sequential updating through projections.

Ideological and Political Exploration for Probability Formulas

Probability theory is an important component of data science, and probability formulas are a key means of analyzing data, occupying a pivotal position in the era of big data. In the context of domestic curriculum ideological and political education, this paper explores the ideological and political elements contained in several probability formulas, seeks the integration of probability formulas and ideological and political education, and makes analogies between the principles, advantages, and applications of these formulas and some ideological and political contents. It not only visualizes abstract probability formulas, but also helps students establish correct worldviews, life values, and moral values. This paper attempts to improve the teaching quality of the courses related to probability and cultivate students’ sense of social responsibility through the ideological and political exploration of these probability formulas.

«Theory of everything»: types of information and the theory of interpretation in «Philosophy of chance» by Stanisław Lem

The article is focused on the S. Lem’s perspective of the literary work as a product of the reader’s apprehension of the text in the culture discourse. The aim of the research is to describe the Lem’s view on the structure of the text; Lem’s encounter as a writer, a reader, and a philosopher is used to state the conditions and characteristics of the text interpretation. By the means of multiple introspection the author of “Philosophy of chance” devotes his research to the explanation of his model of the literary work as a product of the social-cultural existence. Moreover, the author’s aim is to state the characteristics of the text creation and interpretation. Therefore, the research shows a new, “cybernetic” approach to the text interpretation based on the comparison between the text structural characteristics and the rules of the mathematical theory of probability and the theory of information. The futurological view lies in the hypothesis about the text as a “self-learning program” (also known as a neuronet) which can transform due to the developing conditions of the interpretation process. As a matter of fact, S. Lem highlights two types of information: selective and structural. Both types might be identified in the text, but that depends on the reader, his encounter and expertise.

Random Walks and Lorentz Processes

Random walks and Lorentz processes serve as fundamental models for Brownian motion. The study of random walks is a favorite object of probability theory, whereas that of Lorentz processes belongs to the theory of hyperbolic dynamical systems. Here we first present an example where the method based on the probabilistic approach led to new results for the Lorentz process: concretely, the recurrence of the planar periodic Lorentz process with a finite horizon. Afterwards, an unsolved problem—related to a 1981 question of Sinai on locally perturbed periodic Lorentz processes—is formulated as an analogous problem in the language of random walks.