Probability Law Research Articles

Research on the Teaching Design of Deep Learning in High School Mathematics Based on Problem Solving — Taking the Law of Total Probability as an Example

With the rapid development of information technology and artificial intelligence, deep learning has gradually become a major research focus in the field of education. This paper designs an innovative teaching plan for high school mathematics based on problem-solving, using the law of total probability as an example. It is widely agreed that deep learning requires students to face challenging problems and understand the knowledge in the process of solving them. Studies show that problem-oriented teaching strategies can stimulate students’ mathematical thinking and promote deep understanding of concepts and formulas. Through case analysis, this research proposes an effective classroom approach for deep learning in teaching the law of total probability, aiming to improve students' mathematical competency.

Tail index estimation for discrete heavy-tailed distributions with application to statistical inference for regular markov chains

Abstract It is the purpose of this paper to investigate the issue of estimating the regularity index $$\beta >0$$ β > 0 of a discrete heavy-tailed r.v. S, i.e. a r.v. S valued in $$\mathbb {N}^*$$ N ∗ such that $$\mathbb {P}(S>n)=L(n)\cdot n^{-\beta }$$ P ( S > n ) = L ( n ) · n - β for all $$n\ge 1$$ n ≥ 1 , where $$L:\mathbb {R}^*_+\rightarrow \mathbb {R}_+$$ L : R + ∗ → R + is a slowly varying function. Such discrete probability laws, referred to as generalized Zipf’s laws sometimes, are commonly used to model rank-size distributions after a preliminary range segmentation in a wide variety of areas such as e.g. quantitative linguistics, social sciences or information theory. As a first go, we consider the situation where inference is based on independent copies $$S_1,\; \ldots ,\; S_n$$ S 1 , … , S n of the generic variable S. The estimator $$\widehat{\beta }$$ β ^ we propose can be derived by means of a suitable reformulation of the regularly varying condition, replacing S’s survivor function by its empirical counterpart. Under mild assumptions, a non-asymptotic bound for the deviation between $$\widehat{\beta }$$ β ^ and $$\beta $$ β is established, as well as limit results (consistency and asymptotic normality). Beyond the i.i.d. case, the inference method proposed is extended to the estimation of the regularity index of a regenerative $$\beta $$ β -null-recurrent Markov chain. Since the parameter $$\beta $$ β can be then viewed as the tail index of the (regularly varying) distribution of the return time of the chain X to any (pseudo-) regenerative set, in this case, the estimator is constructed from the successive regeneration times. Because the durations between consecutive regeneration times are asymptotically independent, we can prove that the consistency of the estimator promoted is preserved. In addition to the theoretical analysis carried out, simulation results provide empirical evidence of the relevance of the inference technique proposed.

Law of Total Probability of Aftershocks in Earthquake Insurance

Abstract. Seismic hazard expressed in annual rate of exceedance of a peak ground acceleration traditionally refers to mainshock. A similar seismic hazard, APSHA, has been adopted for aftershock probabilistic seismic hazard. Probabilistic seismic hazard assessment (PSHA) refers to a homogeneous Poisson process to describe mainshock while APSHA models aftershock occurrence as a nonhomogeneous Poisson process whose rate modeled as Omori law. It shown that the combination of PSHA and APSHA results seismic hazard for mainshock-aftershock seismic sequence/cluster (SPSHA/CPSHA). This study shows how to combine results of APSHA and PSHA and proposes a method for earthquake insurance. The study was carried out for West Java region with 206 occurrences consist of 11 clusters. One cluster with 74 aftershocks was chosen for further study. The parameters of SPSHA/CPSHA was estimated using maximum likelihood. The results of SPSHA/CPSHA combined with damage probability matrix (DPM) yields an expected annual damage ratio (EADR) as an indicator of earthquake insurance. The proposed method in this study can be used as a method for computing earthquake insurance premium. Due to limited data further study is needed to obtain accurate and reasonable results.

Rank-Based Family of Probability Laws for Testing Homogeneity of Variable Grouping

In order to test within-group homogeneity for numerical or ordinal variable groupings, we have introduced a family of discrete probability distributions, related to the Gini mean difference, that we now study in a deeper way. A member of such a family is the law of a statistic that operates on the ranks of the values of the random variables by considering the sums of the inter-subgroups ranks of the variable grouping. Being so, a law of the family depends on several parameters such as the cardinal of the group of variables, the number of subgroups of the grouping of variables, and the cardinals of the subgroups of the grouping. The exact distribution of a law of the family faces computational challenges even for moderate values of the cardinal of the whole set of variables. Motivated by this challenge, we show that an asymptotic result allowing approximate quantile values is not possible based on the hypothesis observed in particular cases. Consequently, we propose two methodologies to deal with finite approximations for large values of the parameters. We address, in some particular cases, the quality of the distributional approximation provided by a possible finite approximation. With the purpose of illustrating the usefulness of the grouping laws, we present an application to an example of within-group homogeneity grouping analysis to a grouping originated from a clustering technique applied to cocoa breeding experiment data. The analysis brings to light the homogeneity of production output variables in one specific type of soil.

Zero-Sum Mean-Field Dynkin Games: Characterization and Convergence

We introduce a zero-sum game problem of mean-field type as an extension of the classical zero-sum Dynkin game problem to the case where the payoff processes might depend on the value of the game and its probability law. We establish sufficient conditions under which such a game admits a value and a saddle point. Furthermore, we provide a characterization of the value of the game in terms of a specific class of doubly reflected backward stochastic differential equations of mean-field type, for which we derive an existence and uniqueness result. We then introduce a corresponding system of weakly interacting zero-sum Dynkin games and show its well-posedness. Finally, we provide a propagation of chaos result for the value of the zero-sum mean-field Dynkin game.

Bayesianism, Conditional Probability and Laplace Law of Succession in Quantum Mechanics

Bayesianism, Conditional Probability and Laplace Law of Succession in Quantum Mechanics

Positive Lymph Nodes Independently Affect Long-Term Survival After Pancreaticoduodenectomy for Non-Ampullary Duodenal Adenocarcinoma: A Single-Center, Retrospective Analysis.

Background/Objectives: The main treatment for non-ampullary duodenal adenocarcinoma (NDA) is pancreatoduodenectomy (PD) with lymphadenectomy (LN). Several studies have proposed a minimum number of examined lymph nodes (MNELN) to ensure proper staging. This study investigated the impact of nodal parameters-including the pattern of nodal spread-on oncologic outcomes following PD for NDA. Furthermore, we sought to determine the MNELN to ensure reliable detection of nodal involvement. Methods: This was a single-center, retrospective study. Consecutive patients who underwent PD from 2000 to 2019 with a final diagnosis of NDA were retrieved from a prospectively maintained database. The probability of detecting at least one metastatic LN in a node-positive patient was assessed using a model based on the binomial probability law. Results: A total of 70 patients met the inclusion criteria. The median number of ELNs was 35 (22-43, IQR). Thirty-six patients (51%) had at least one PLN. A node-positive disease was associated with adverse pathologic features, including high tumor grade and perineural and peripancreatic fat invasion. This translated into a greater recurrence rate (p < 0.001). The MNELN yielding a 95% probability of detecting at least one metastatic node in a node-positive patient was 25. After a median follow-up of 73 months, the median recurrence-free survival (RFS) was 33 months (95% CI 13-97), and the overall survival (OS) was 41 months (95% CI 17-96). The LN ratio, tumor grade, and metastases at stations 8 and 12 were independently associated with OS (p < 0.05). Conclusions: Nodal metastases are common among patients with NDA and have a considerable impact on long-term survival. Stations 8 and 12 were associated with OS. Therefore, an adequate lymphadenectomy, possibly including stations 8 and 12, is recommended in patients with NDA.

A universal power law optimizes energy and representation fidelity in visual adaptation.

Sensory systems continuously adapt their responses based on the probability of encountering a given stimulus. In the mouse primary visual cortex (V1), the average population response is a power law of the prior probability of stimuli in the environment. For a given stimulus type (e.g., oriented gratings), the power law's exponent is invariant to changes in statistical environments, enabling predictions of average population responses to new environments. Here, we aim to provide a normative explanation for the power law behavior. We develop an efficient coding model where neurons adjust their firing rates through multi-objective optimization, hypothesizing that the neural population adapts to enhance stimulus detection and discrimination while reducing overall neural activity. We show that a power law that matches the one observed experimentally can emerge from our model. We interpret the exponent as reflecting a balance between energy efficiency and representational fidelity in adaptation. Furthermore, we account for the invariance of the power law's exponent across environmental changes by linking it to the dependence of tuning curve modulation on stimulus probability. Finally, we explain that variations in the exponent with different stimulus types (e.g., natural movies) result from changes in the minimal distances between neural representations, in agreement with experimental findings. We conclude that a universal power law of adaptation can be explained as a trade-off between representation fidelity and energy cost.

A Non-Parametric Test for a Two-Way Analysis of Variance

The methodology carried out in this work is based on non-parametric inference. The problem is framed as a regression analysis, and the solution is derived using the permutation approach. The proposed test does not rely on the assumption that the distribution of the response follows a specific family of probability laws, unlike other parametric approaches. This makes the test powerful, particularly when the typical assumptions of parametric approaches, such as the normality of data, are not satisfied and parametric tests are not reliable. Furthermore, this method is more flexible and robust with respect to parametric tests. A permutation test on the goodness-of-fit of a multiple regression model is applied. Hence, proposed solution consists of the application of permutation tests on the significance of the single coefficients and then a combined permutation test (CPT) to solve the overall goodness-of-fit testing problem. Furthermore, a Monte Carlo simulation study was performed to evaluate the power of the previously mentioned permutation approach, comparing it with the conventional parametric F-test for ANOVA and the bootstrap combined test, both commonly discussed in the literature on this statistical problem. Finally, the proposed non-parametric test was applied to real-world data to investigate the impact of age and smoking habits on medical insurance costs in the USA. The findings suggest that smoking and being at least 50 years old significantly contribute to increased medical insurance costs.

Optimization Problem for Probabilistic Time Intervals of Quasi-Deterministic Output and Self-Similar Input Data Packet Flow in Telecommunication Networks

Introduction. When managing traffic at the packet level in modern telecommunication networks, it is proposed to use methods that transform a self-similar stochastic packet flow into a quasi-deterministic one. To do this, it is required to apply complex probabilistic laws of distribution of self-similar flows. From the literature, methods of balancing the network load are known, which, with the problem indicated above, contribute to increasing the efficiency of telecommunication systems. However, there is no strictly mathematical solution to find out the optimal probabilistic characteristics of the output flow, based on the input flow. The presented research is intended to fill this gap. Its objective is to create a method for determining the optimal probabilistic characteristics of the packet flow, using the minimum value of the proximity measure of the self-similar input and quasi-deterministic output flows.Materials and Methods. To solve the research problem, the parameters of the output flow distribution were selected so that the approximation function was close to 𝛿𝛿-function. The Kullback-Leibler divergence was used as a proximity measure of the input and output distributions of time intervals. Methods of set theory, metric spaces, multidimensional optimization, and teletraffic were used. The solution algorithm included minimization of the Kullback-Leibler divergence and the limit passage to 𝛿𝛿-function.Results. A probability distribution is shown — an approximation of 𝛿𝛿-function, which maintains the equality of time intervals of a quasi-deterministic output packet flow. A method for transforming a self-similar input flow into a quasideterministic output flow is presented. The Kullback–Leibler divergence was used as a measure of their proximity. The minimum of the Kullback-Leibler divergence between the input and output flows with a normal distribution was achieved in the case of equality of the mathematical expectations of these flows. Using the passage to the limit, it has been established that time interval T between packets of the quasi-deterministic output flow must be equal to the mathematical expectation of the time intervals between packets of the input self-similar flow. To obtain a quasi-deterministic flow, the passage to the limit is performed for the found value of the mathematical expectation at σ → 0.Discussion and Conclusion. The application of this method will reduce the negative impact of self-similarity of network traffic on the efficiency of the telecommunication network. The use of quasi-deterministic flows makes it possible to predict the load of network resources, which can be the basis for improving the quality of user service. Two difficulties associated with calculations and practical implementation of the solution are eliminated. Firstly, it is difficult to use the delta function as a function of the output flow distribution density. Secondly, there are no ideal deterministic flows in the operation of telecommunication networks. The proposed method has great potential in the design and optimization of communication networks.

Faster algorithms for the alignment of sparse correlated Erdős–Rényi random graphs

Abstract The correlated Erdős–Rényi random graph ensemble is a probability law on pairs of graphs with n vertices, parametrized by their average degree λ and their correlation coefficient s. It can be used as a benchmark for the graph alignment problem, in which the labels of the vertices of one of the graphs are reshuffled by an unknown permutation; the goal is to infer this permutation and thus properly match the pairs of vertices in both graphs. A series of recent works has unveiled the role of Otter’s constant α (that controls the exponential rate of growth of the number of unlabeled rooted trees as a function of their sizes) in this problem: for s &gt; α and large enough λ, it is possible to recover in a time polynomial in n a positive fraction of the hidden permutation. The exponent of this polynomial growth is, however, quite large and depends on the other parameters, which limits the range of applications of the algorithm. In this work, we present a family of faster algorithms for this task, show through numerical simulations that their accuracy is only slightly reduced with respect to the original one and conjecture that they undergo, in the large λ limit, phase transitions at modified Otter’s thresholds α ^ &gt; α , with α ^ related to the enumeration of a restricted family of trees.

Monitoring and Studying the Behavior of Metals in an Industrial Wastewater Treatment Plant in Italy

Heavy metals represent a significant hazard in textile wastewater, posing a considerable risk to both the ecosystem and human health. The objective of this study was to analyze the removal efficiency of specific heavy metals in a large wastewater treatment plant (WWTP) located in Prato (Tuscany, Italy), where the main Italian textile district is based. To achieve this, the mass balance calculation approach was employed. Therefore, two monitoring campaigns were conducted, collecting wastewater and sludge samples in some specific sections of the WWTP. The concentrations of Pb, Cd, Ni, As, and Sn were consistently below the detection limits. A good removal efficiency was determined for Zn, Cu, Ba, Crtot, and Sb, in the range of 37–79%. These metals are predominantly present in particulate form, facilitating their removal through sedimentation. Conversely, boron is largely present in the dissolved phase, resulting in its complete release through the treated effluent. Subsequently, an excellent linear correlation was identified between the input load and the contaminant load removed. This demonstrated that the plant’s efficiency remains unaffected by an increase in the input load at the observed contaminant concentrations. Finally, a probability law was identified that demonstrates an excellent degree of approximation in representing inlet metal concentrations. The findings of this study indicate that the treatment systems employed by the WWTP are capable of effectively removing heavy metals.

The Bijection Property in the Law of Total Probability and Its Application in Communication Theory

The Bijection Property in the Law of Total Probability and Its Application in Communication Theory

Bayesian inference: more than Bayes’s theorem

Bayesian inference gets its name from Bayes’s theorem, expressing posterior probabilities for hypotheses about a data generating process as the (normalized) product of prior probabilities and a likelihood function. But Bayesian inference uses all of probability theory, not just Bayes’s theorem. Many hypotheses of scientific interest are composite hypotheses, with the strength of evidence for the hypothesis dependent on knowledge about auxiliary factors, such as the values of nuisance parameters (e.g., uncertain background rates or calibration factors). Many important capabilities of Bayesian methods arise from use of the law of total probability, which instructs analysts to compute probabilities for composite hypotheses by marginalization over auxiliary factors. This tutorial targets relative newcomers to Bayesian inference, aiming to complement tutorials that focus on Bayes’s theorem and how priors modulate likelihoods. The emphasis here is on marginalization over parameter spaces—both how it is the foundation for important capabilities, and how it may motivate caution when parameter spaces are large. Topics covered include the difference between likelihood and probability, understanding the impact of priors beyond merely shifting the maximum likelihood estimate, and the role of marginalization in accounting for uncertainty in nuisance parameters, systematic error, and model misspecification.

DẠY HỌC XÁC SUẤT - THỐNG KÊ GẮN VỚI THỰC TIỄN NGHỀ NGHIỆP CHO SINH VIÊN TRƯỜNG ĐẠI HỌC Y DƯỢC THÁI BÌNH

Based on the teaching process in the direction of developing learners’ capability and research results on the Theory of Realistic Mathematics Education, the paper proposes an educational process for using research results in Medicine and Pharmacy to design content for teaching probability and statistics at Thai Binh University of Medicine and Pharmacy, including 5 steps: (1) Finding “realistic situations” from the field of medicine and pharmacy that are suitable for students; (2) Organizing activities for learners to perform learning tasks; (3) Formalizing; (4) Applying; (5) Assessing and Self-assessing. The pedagogical experiment with the content "The law of Total probability" was conducted with the Preventive Healthcare major students at Thai Binh University of Medicine and Pharmacy in the academic year 2023 - 2024. The results showed that the proposed measure improves interest in learning and links probability and statistics knowledge with professional practice.

Tractable optimal experimental design using transport maps*

Abstract We present a flexible method for computing Bayesian optimal experimental designs (BOEDs) for inverse problems with intractable posteriors. The approach is applicable to a wide range of BOED problems and can accommodate various optimality criteria, prior distributions and noise models. The key to our approach is the construction of a transport-map-based surrogate to the joint probability law of the design, observational and inference random variables. This order-preserving transport map is constructed using tensor trains and can be used to efficiently sample from (and evaluate approximate densities of) conditional distributions that are required in the evaluation of many commonly-used optimality criteria. The algorithm is also extended to sequential data acquisition problems, where experiments can be performed in sequence to update the state of knowledge about the unknown parameters. The sequential BOED problem is made computationally feasible by preconditioning the approximation of the joint density at the current stage using transport maps constructed at previous stages. The flexibility of our approach in finding optimal designs is illustrated with some numerical examples inspired by disease modeling and the reconstruction of subsurface structures in aquifers.

THE GENERALIZED BIVARIATE POISSON DISTRIBUTION ACCORDING TO BERKHOUT AND PLUG

In this article, we will construct a new bivariate Poisson distribution through the bivariate law of probabilities of causes highlighted by Bidounga et al. in [\ref{BP52}]. This law generalise the bivariate Poisson distribution according to Berkhout and Plug [\ref{BP50}].And finally we simulated the data.

Book Review of Probability Theory:The Logic of Science

E. T. Jaynes' Probability Theory: The Logic of Science is a classical work that delves deeply into the role of probability as the foundation of scientific reasoning. The author extends probability as a continuation of logic, demonstrating how Bayesian inference can unify the treatment of uncertainty across various scientific disciplines. The book covers the fundamental laws of classical probability while thoroughly exploring its extensive applications in fields such as statistical physics, economics, and data analysis. With a writing style. that balances theoretical rigor and practical insights, Jaynes offers readers a fresh perspective on understanding probability, making the book an essential reference for both scientific research and practical application.

An Information-Geometric Formulation of Pattern Separation and Evaluation of Existing Indices.

Pattern separation is a computational process by which dissimilar neural patterns are generated from similar input patterns. We present an information-geometric formulation of pattern separation, where a pattern separator is modeled as a family of statistical distributions on a manifold. Such a manifold maps an input (i.e., coordinates) to a probability distribution that generates firing patterns. Pattern separation occurs when small coordinate changes result in large distances between samples from the corresponding distributions. Under this formulation, we implement a two-neuron system whose probability law forms a three-dimensional manifold with mutually orthogonal coordinates representing the neurons' marginal and correlational firing rates. We use this highly controlled system to examine the behavior of spike train similarity indices commonly used in pattern separation research. We find that all indices (except scaling factor) are sensitive to relative differences in marginal firing rates, but no index adequately captures differences in spike trains that result from altering the correlation in activity between the two neurons. That is, existing pattern separation metrics appear (A) sensitive to patterns that are encoded by different neurons but (B) insensitive to patterns that differ only in relative spike timing (e.g., synchrony between neurons in the ensemble).

A computational approach to extreme values and related hitting probabilities in level-dependent quasi-birth–death processes

A computational approach to extreme values and related hitting probabilities in level-dependent quasi-birth–death processes