Problem In Probability Theory Research Articles

Asymptotic Probability Expansions for Random Elements in a Hilbert space

In this article, we approach a class of problems in probability theory, namely, the asymptotic expansion of probability. We consider an independent, identically distributed, and normalized stochastic process in a separable Hilbert space H, and associate it with the normalized partial sum.As a result, we built on the ball with a fixed center asymptotic expansion of non-uniform probabilities; our conditions on the moments are minimal, and the dependency of estimates on the covariance operator is expressed with the terms of the eigenvalue series. Likewise, the covariance operators of the random elements do not coincide. In the open ball set with fixed center a and radius , we estimate the optimal result of the Berry-Esseen type of the remainder, and the terms of the probability by the Fourier method.

A Basis for AGM Revision in Bayesian Probability Revision

In standard Bayesian probability revision, the adoption of full beliefs (propositions with probability 1) is irreversible. Once an agent has full belief in a proposition, no subsequent revision can remove that belief. This is an unrealistic feature, and it also makes probability revision incompatible with belief change theory, which focuses on how the set of full beliefs is modified through both additions and retractions. This problem in probability theory can be solved in a model that (i) lets the codomain of the probability function be a hyperreal-valued rather than the real-valued closed interval [0, 1], and (ii) identifies the full beliefs as the propositions whose probability is either 1 or infinitesimally smaller than 1. In this model, changes in the probability function will result in changes in the set of full beliefs (belief set), which constitutes a submodel that can be conceived as the “tip of the iceberg” within the larger model that also contains beliefs on lower levels of probability. The patterns of change in the set of full beliefs in this modified Bayesian model coincides with the corresponding pattern in a slightly modified version of AGM revision, which is commonly conceived as the gold standard of (dichotomous) belief change. The modification only concerns the marginal case of revision by an inconsistent input sentence. These results show that probability revision and dichotomous belief change can be unified in one and the same framework, or – if we so wish – that belief change theory can be subsumed under a modified version of probability revision that allows for iterated change and for the removal of full beliefs.

Polynomials and number sets associated with the probability distribution of the hyperbolic cosine type for even values of the parameter

The polynomials used in the formation of the probability distribution density function of the hyperbolic cosine type are investigated. Earlier, on the basis of a hyperbolic cosine distribution, the author obtained numerical sets, among which not only new ones, but also, for example, the triangle of Stirling numbers, the triangle of the coefficients of Bessel polynomials, sequences of coefficients in the expansion of various functions, etc. In this paper, depending on the natural parameter m and the real distribution parameter β , a new class of polynomials is obtained. For even and odd m , the polynomials are constructed using similar, but different formulas. The article presents polynomials for even values m . Structurally, polynomials consist of quadratic factors. The coefficients of the polynomials, ordered by m , form numerical triangles depending on β . Some relations are found between the coefficients. From the numerical triangles, a set of numerical sequences is obtained, which for integers β are integers. Also, polynomials with respect to x turn out to be polynomials with respect to β . With this interpretation the variable acts as a parameter. New numerical triangles and sequences for different x were found. The overwhelming majority of the obtained numerical sequences are new. The class of polynomials arising from problems of probability theory indicates the possibility of applying the results.

Fundamentals of metric theory of real numbers in their $\overline{Q_3}$-representation

In the paper we were studied encoding of fractional part of a real number with an infinite alphabet (set of digits) coinciding with the set of non-negative integers. The geometry of this encoding is generated by $Q_3$-representation of real numbers, which is a generalization of the classical ternary representation. The new representation has infinite alphabet, zero surfeit and can be efficiently used for specifying mathematical objects with fractal properties.
 We have been studied the functions that store the "tails" of $\overline{Q_3}$-representation of numbers and the set of such functions,some metric problems and some problems of probability theory are connected with $\overline{Q_3}$-representation.

The model-specific Markov embedding problem for symmetric group-based models

We study model embeddability, which is a variation of the famous embedding problem in probability theory, when apart from the requirement that the Markov matrix is the matrix exponential of a rate matrix, we additionally ask that the rate matrix follows the model structure. We provide a characterisation of model embeddable Markov matrices corresponding to symmetric group-based phylogenetic models. In particular, we provide necessary and sufficient conditions in terms of the eigenvalues of symmetric group-based matrices. To showcase our main result on model embeddability, we provide an application to hachimoji models, which are eight-state models for synthetic DNA. Moreover, our main result on model embeddability enables us to compute the volume of the set of model embeddable Markov matrices relative to the volume of other relevant sets of Markov matrices within the model.

Accurately approximating extreme value statistics

We consider the extreme value statistics of N independent and identically distributed random variables, which is a classic problem in probability theory. When N → ∞, fluctuations around the maximum of the variables are described by the Fisher–Tippett–Gnedenko theorem, which states that the distribution of maxima converges to one out of three limiting forms. Among these is the Gumbel distribution, for which the convergence rate with N is of a logarithmic nature. Here, we present a theory that allows one to use the Gumbel limit to accurately approximate the exact extreme value distribution. We do so by representing the scale and width parameters as power series, and by a transformation of the underlying distribution. We consider functional corrections to the Gumbel limit as well, showing they are obtainable via Taylor expansion. Our method also improves the description of large deviations from the mean extreme value. Additionally, it helps to characterize the extreme value statistics when the underlying distribution is unknown, for example when fitting experimental data.

Some applications of the GRAN1 to analyze two-dimensional continuous probability distributions

The article considers the use of GRAN1 software in the analysis of two-dimensional continuous probability distributions. The learning process can be provided by freely distributable software, through the use of cloud technologies, in particular software from the GRAN package. However, the use of any technology in the educational process, including modern information and communication technologies, as well as the content of education, must be pedagogically balanced, which will avoid any negative consequences for the formation of the future member of society, his mental and physical development. Attention is drawn to the fact that children are increasingly looking for entertainment, using mobile devices, sharing new products with their peers and spending time together, playing a variety of online games, and parents cannot control this process. At the same time, teachers conduct lessons using mobile devices to draw attention to the subject, they are teaching. In the process of using computer technology, students can form a special type of thinking, the so-called “clip consciousness”, which leads to fragmentary thinking and the irreversibility of changes in consciousness. Therefore, the problem of excessive use of mobile devices by children requires constant analysis and coverage. The article discusses some examples of using the GRAN1 software to solve problems in probability theory. Examples of approximate calculation of the value of the double integral are given. Tables of values of the corresponding integrals are described. The use of modern information and communication technologies during mathematics lessons is considered.

СТОХАСТИЧНІ ЗАДАЧІ І ПРИКЛАДНА СПРЯМОВАНІСТЬ У НАВЧАННІ МАТЕМАТИКИ

The article deals with some aspects of the study of the discipline “Probability theory and mathematical statistics” by the students of economic specialties through the solution of practical exercises. The main aim of learning the course is to form the skills to apply the knowledge gained to the tasks in economics. The concept of teaching probability theory and mathematical statistics is a process in which stochastic concepts and ideas serve as a mathematical apparatus for solving specific problems. Nowadays it is traditional to acquaint students with certain sections of applied mathematics, including probability theory and mathematical statistics, as purely abstract theories. However, as the best we consider another approach, according to which “a more adequate solution will be to acquaint the students with the methods of mathematical models construction”. In solving the problems of an applied nature, students get an idea of the necessity and universality of mathematics and its methods. The value of stochastic problems is determined predominantly not by the apparatus used in the process of their solution, but by the ability to demonstrate the process of usage of mathematics in solution of non-mathematical problems. It is shown how, with the help of applied tasks, to familiarize students with real examples of application of stochastic ideas and methods, as well as to make it possible to organize specific activities necessary in the process of application of mathematics. The student, researching a mathematical problem, formulates different questions and problems, then “transforms” them into the notions of mathematics, in order to solve them by mathematical methods, and then adapt the solution to the real problem, which was set at the beginning of the learning activity. This process is a process of constructing a mathematical (probabilistic) model of a real situation, which can be considered a mathematical activity in a broad sense. Among the traditional stochastic problems there are many typically mathematical (intramodel) tasks of such a kind, which were formulated by means of non-mathematical terms. It is necessary to note that the real problems of an applied nature are rare in mathematics because the stage of formalization (construction of a mathematical model of a non-mathematical situation) requires to have a deep knowledge and mathematical culture. This fact generated the problem of selection of tasks of applied nature that can be used in teaching. The paper presents a number of examples where students are shown how, with the help of some modification, a number of traditional problems of probability theory (formulated in the language of nonmathematical terms) can be developed into the tasks of applied nature. Expanding the range of such tasks during the study of mathematics would have a positive effect on students’ attitudes to this discipline and would increase their motivation to learn. Besides, the role of stochastic issues in mathematics and general education would become more multifaceted. For the university teachers of mathematics working with the students of economic specialties the most important thing is the formation of students’ mathematical skills to use the mathematical apparatus in their future professional activities.

Embeddability of real and positive operators

Embedding discrete Markov chains into continuous ones is a famous open problem in probability theory with many applications. Inspired by recent progress, we study the closely related questions of embeddability of real and positive operators into real or positive -semigroups, respectively, on finite and infinite-dimensional separable sequence spaces. For the real case we give both sufficient and necessary conditions for embeddability. For positive operators we present necessary conditions for positive embeddability including a full description for the -case. Moreover, we show that real embeddability is topologically typical for real contractions on .

The Coupon Collection Behavior in Human Reproduction

There is evidence in humans for genetic influences on the probability (Pboy) that a birth yields a boy instead of a girl [1-6], suggesting a potential variation of Pboy among families. To quantify this variation, we analyze the survey data from over 300,000UK Biobank participants primarily born between 1940 and 1970 [7]. Surprisingly, the proportion of male children in a family, or sex ratio (SR), has a significantly smaller among-family variation than expected under a uniform Pboy. We propose that this phenomenon results from reproductive behaviors reflecting a preference for having children of both sexes, much like the coupon collector's problem in probability theory where collecting a complete set of distinct coupons is considered a win. We find that the observed deficit in SR variation is explainable by 3.3% of "coupon-collecting" families. Consistently, significantly more families than expected have all children of the same sex except for the child born last. This trend is more pronounced in the late than the early half of the families in the data, suggesting an increasing popularity of this behavior. Analysis of a Dutch genealogical dataset spanning the past 4 centuries reveals higher-than-expected SR variations over much of the history; only after 1940 did the SR variation drop below the expectation. We conclude that a significant fraction of couples now exhibit the coupon collection behavior in reproduction such that SR is more homogeneous among families than expected by chance.

On biased random walks, corrupted intervals, and learning under adversarial design

We tackle some fundamental problems in probability theory on corrupted random processes on the integer line. We analyze when a biased random walk is expected to reach its bottommost point and when intervals of integer points can be detected under a natural model of noise. We apply these results to problems in learning thresholds and intervals under a new model for learning under adversarial design.

Where Does Satiation Come From?

We introduce a simple – yet novel – queuing system composed of particle inputs, receptors, and response outputs. The system is a natural extension of the occupancy problem in probability theory. A possible interpretation of the model is to have the response output be the utility an individual obtains from input consumption, subject to the biological processing constraints. We derive the response function of the system, and show it can be closely approximated by a discounted utility model with a satiation state. Next, we show that the there is an upper bound to the utility rate the individual can experience in stationary conditions. Finally, we derive the temporal distribution of inputs to maximizes the total response under a budget constraint.

Estimation of Probability Distribution and Its Application in Bayesian Classification and Maximum Likelihood Regression

Nonparametric estimation of cumulative distribution function and probability density function of continuous random variables is a basic and central problem in probability theory and statistics. Although many methods such as kernel density estimation have been presented, it is still quite a challenging problem to be addressed to researchers. In this paper, we proposed a new method of spline regression, in which the spline function could consist of totally different types of functions for each segment with the result of Monte Carlo simulation. Based on the new spline regression, a new method to estimate the distribution and density function was provided, which showed significant advantages over the existing methods in the numerical experiments. Finally, the density function estimation of high dimensional random variables was discussed. It has shown the potential to apply the method in classification and regression models.

Convex duality in nonlinear optimal transport

This article studies problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of a large class of related problems in probability theory and allows for generalizations of the classical problem formulations. General results on convex duality yield dual problems and optimality conditions for these problems. When the objective takes the form of a convex integral functional, we obtain more explicit optimality conditions and establish the existence of solutions for a relaxed formulation of the problem. This covers, in particular, the mass transportation problem and its nonlinear generalizations.

Asymptotic Expansion in Multiplicative Form in the Central Limit Theorem in the Case of Gamma Distribution

Study of estimation of accuracy of approximations in the Central limit theorem (CLT) is one of the known problems in probability theory. The main result here is the estimate of the theorem of Berry — Esseen. Its low accuracy is well known. So this theorem guarantees accuracy of approximation 103 in the CLT only if the number of summands in the normed sum is greater than 160 000. Therefore, increasing the accuracy of the approximations in the CLT is an actual task. In particular, for this purpose are used asymptotic expansions in the Central limit theorem. As a rule, asymptotic expansions have additive form. Although it is possible to construct expansions in the multiplicative form. So V.M. Kalinin in [3] received the multiplicative form of the asymptotic expansions. However, he constructed asymptotic expansions for probability distributions (multinomial, Poisson, Student’s t-distribution). So very naturally the question arises: how to build multiplicative expansions in CLT? Secondly, what are the forms of decompositions in CLT in terms of accuracy approximations are better: additive or multiplicative? This paper proposes new asymptotic expansions in the central limit theorem which permit us to approximate distributions of normalized sums of independent gamma random variables with explicit estimates of the approximation accuracy and comparing them with expansions in terms of Chebyshev — Hermite polynomials. New asymptotic expansions is presented in the following theorem. Comparing multiplicative asymptotic expansion from theorem 1 with the additive asymptotic expansion from [5], we obtain that multiplicative asymptotic expansion of the density of normalized the sums in the case of gamma distribution give a much greater accuracy numerical calculations are compared with asymptotic additive expansion provided a much smaller number of calculations. The author would like to thank Vladimir Senatov for setting the task and paying attention to this work.

Natural selection and the reference grain problem

Over the last 20 years, the concept of natural selection has been highly debated in the philosophy of biology. Yet, most discussions on this topic have focused on the questions of whether natural selection is a causal process and whether it can be distinguished from drift. In this paper, I identify another sort of problem with respect to natural selection. I show that, in so far as a classical definition of fitness includes the transmission of a type between generations as part of the definition, it seems difficult to see how the fitness of an entity, following this definition, could be description independent. In fact, I show that by including type transmission as part of the definition of fitness, changing the grain at which the type of an entity is described can change the fitness of that entity. If fitness is not grain-of-description independent, this further propagates to the process of natural selection itself. I call this problem the ‘reference grain problem’. I show that it can be linked to the reference class problem in probability theory. I tentatively propose two solutions to it.

Influence of Randomness During the Interpretation of Results From Single-Event Experiments on SRAMs

After having carried out radiation experiments on memories, the detected bitflips must be classified into single bit upsets and multiple events to calculate the cross sections of different phenomena. There are some accepted procedures to determine if two bitflips are related. However, if there are enough bitflips, it is possible that unrelated pairs of errors appear in nearby cells and they are erroneously taken as a multiple event. In this paper, radiation experiments are studied as a special case of the urn-and-balls problem in probability theory to estimate how the measured multiple-event cross sections must be corrected to remove the overestimation due to the false events.

REVERSAL FUNCTIONS AND PROBABILITY DISTRIBUTION OF THE PANDOM CROSS-FUNCTIONAL FROM NORMAL MARKOV PROCESS

The process, which has the properties of stationary, normality and markovity, is considered. For a given time interval, an energy functional and a cross-functional are studied in this article. In analytic consideration of the problems of probability theory and mathematical statistics, the assumption is widespread that the problem in question has been resolved if the characteristic (generating) function is constructed. However, the operation of the inverse Laplace transform causes the computational difficulties. As a numerical procedure, the inverse Laplace transform is characterized by instability, the degree of which increases with the transformation parameter. An approach based on the application of the reverse functions was proposed and used, which made it possible to obtain an analytical expression for the generating function of the distribution of random values of the cross-functional in this article. The analysis of the statistical properties of the cross functional is carried out. The density of probability distribution and the integral distribution law are obtained numerically using the inverse Laplace transform for the selected values of the observation time , the decrement of the process and its intensity . The dependences of the density and distribution functions for given values of the parameters of functionals are given. It follows from the calculations that an increase in the parameter leads to the expansion of the values of the functional in the peripheral regions of large deviations. Reducing the parameter leads to the localization of the values of the cross-functional in the fluctuation domain . The density is symmetric with respect . It has a unique maximum, two points of inflection, and an exponential asymptotic behavior on the periphery.

Multivariate Analogs of Classical Univariate Discrete Distributions and Their Properties

Some discrete distributions such as Bernoulli, binomial, geometric, negative binomial, Poisson, Polya–Aeppli, and others play an important role in applied problems of probability theory and mathematical statistics. We propose a variant of a multivariate distribution whose components have a given univariate discrete distribution. In fact we consider some very general variant of the so-called reduction method. We find the explicit form of the mass function and generating function of such distribution and study their properties. We prove that our construction is unique in natural exponential families of distributions. Our results are the generalization and unification of many results of other authors.

On Zadeh's problem in probability theory

ABSTRACTWe pay tribute to Lotfi Zadeh, a long-term member of the editorial board of this journal, by drawing attention to his challenges to classical probability theory. In particular, we consider the following problem: Probably John is tall. What is the probability that John is short? What is the probability that John is very short? What is the probability that John is not very tall? We discuss possible solutions and general significance of this and similar problems.