Formula For Probability Research Articles

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.

Decoding error probability of random parity-check matrix ensemble over the erasure channel

AbstractIn this paper we carry out an in-depth study on the average decoding error probability of the random parity-check matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random parity-check matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random parity-check matrix ensemble and the error exponent of the variance, which implies a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random linear code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.

A note on the satisficing policy of the secretary problem

Wu and Li (2022) presented a satisficing policy for the secretary problem (SP). Moreover, Wu and Li (2022) derived asymptotic formulas of the maximum probability of choosing one of the k satisfactory applicants with special cases k = 2 and 3, respectively, in the satisficing policy. “Based on the work of Wu and Li (2022), this paper derives the asymptotic formulas for the probability of selecting one of the k satisfactory applicants and provides the equation for the approximate optimal cutoff value to achieve the asymptotic maximum probability in the general case of k satisfactory applicants under the satisficing policy.” Moreover, a series of computer simulations are conducted to test the theoretical results for k = 4 and 5.

Association Model-Based Intermittent Connection Fault Diagnosis for Controller Area Networks

Controller Area Networks (CANs) play an important role in many safety-critical industrial systems, which places high demands on their reliability performance. However, the intermittent connection (IC) of network cables, a random and transient connectivity problem, is a common but hard troubleshooting fault that can cause network performance degradation, system-level failures, and even safety issues. Therefore, to ensure the reliability of CANs, a fault symptom association model-based IC fault diagnosis method is proposed. Firstly, the symptoms are defined by examining the error records, and the domains of the symptoms are derived to represent the causal relationship between the fault locations and the symptoms. Secondly, the fault probability for each location is calculated by minimizing the difference between the symptom probabilities calculated from the count information and those fitted by the total probability formula. Then, the fault symptom association model is designed to synthesize the causal and the probabilistic diagnostic information. Finally, a model-based maximal contribution diagnosis algorithm is developed to locate the IC faults. Experimental results of three case studies show that the proposed method can accurately and efficiently identify various IC fault location scenarios in networks.

Performance enhancement of NOMA multi-user networks with aerial reconfigurable intelligent surfaces and aerial relay

In this article, we propose a combination of multiple aerial reconfigurable intelligent surfaces (ARISs) with an aerial relay (AR) for improving the performance of multiple users adopting the non-orthogonal multiple access (NOMA) scheme. In particular, two groups of ARIS are utilized, one for aiding ground-to-air (G2A) communication and another one for aiding air-to-ground (A2G) communication. Mathematical formulas of outage probability (OP), throughput, and achievable capacity (AC) of every user in the proposed ARISs-AR-NOMA network are derived over the Nakagami-m channel. The propagation environment recommended for the fifth generation and beyond (5G/B5G) is applied to enhance the practical behavior of the proposed network. To confirm the performance enhancement, we compare the OP, throughput, and AC of the proposed system with the conventional AR-NOMA network (i.e., without ARISs). Numerical illustrations demonstrate the great benefits of the proposed ARISs-AR-NOMA network, its transmit power is dramatically lower than that of the conventional network. Moreover, the throughput and AC of the proposed network are considerably higher than the conventional network’s. Based on the effect of key parameters and network behaviors, various effective solutions are recommended to enhance the performance of the proposed ARISs-AR-NOMA network.

Ruin Probabilities as Recurrence Sequences in a Discrete-Time Risk Process

The theory of linear recurrence sequences is applied to obtain an explicit formula for the ultimate ruin probability in a discrete-time risk process. It is assumed that the claims distribution is arbitrary but has finite support {0,1,…,m+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\{0,1,\\ldots ,m+1\\}}$$\\end{document}, for some integer m≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{m\\ge 1}$$\\end{document}. The method requires finding the zeroes of an m degree polynomial and solving a system of m linear equations. An approximation is derived and some numerical results and plots are provided as examples.

Stochastic mortality model with respect to mixed fractional Poisson process: Calibration and empirical analysis of long-range dependence in actuarial valuation

Recently, many studies have adopted the fractional stochastic mortality process in characterising the long-range dependence (LRD) feature of mortality dynamics, while there are still fewer appropriate non-Gaussian fractional models to describe it. We propose a stochastic mortality process driven by a mixture of Brownian motion and modified fractional Poisson process to capture the LRD of mortality rates. The survival probability under this new stochastic mortality model keeps flexibility and consistency with existing affine-form mortality models, which makes the model convenient in evaluating mortality-linked products under the market-consistent method. The formula of survival probability also considers the historical information from survival data, which enables the model to capture historical health records of lives. The LRD feature is reflected by our proposed model in the empirical analysis, which includes the calibration and prediction of survival curves based on recent generation data in Japan and the UK. Finally, the consequent empirical analysis of annuity pricing illustrates the difference of whether this feature is involved in actuarial valuation.

An Improved Ant Colony Algorithm with Deep Reinforcement Learning for the Robust Multiobjective AGV Routing Problem in Assembly Workshops

Vehicle routing problems (VRPs) are challenging problems. Many variants of the VRP have been proposed. However, few studies on VRP have combined robustness and just-in-time (JIT) requirements with uncertainty. To solve the problem, this paper proposes the just-in-time-based robust multiobjective vehicle routing problem with time windows (JIT-RMOVRPTW) for the assembly workshop. Based on the conflict between uncertain time and JIT requirements, a JIT strategy was proposed. To measure the robustness of the solution, a metric was designed as the objective. Afterwards, a two-stage nondominated sorting ant colony algorithm with deep reinforcement learning (NSACOWDRL) was proposed. In stage I, ACO combines with NSGA-III to obtain the Pareto frontier. Based on the model, a pheromone update strategy and a transfer probability formula were designed. DDQN was introduced as a local search algorithm which trains networks through Pareto solutions to participate in probabilistic selection and nondominated sorting. In stage II, the Pareto frontier was quantified in feasibility by Monte Carlo simulation, and tested by diversity-robust selection based on uniformly distributed weights in the solution space to select robust Pareto solutions that take diversity into account. The effectiveness of NSACOWDRL was demonstrated through comparative experiments with other algorithms on instances. The impact of JIT strategy is analyzed and the effect of networks on the NSACOWDRL is further discussed.

Asymptotics for ruin probabilities of a bidimensional risk model with a random number of delayed claims

This article considers a non-standard bidimensional risk model with constant force of interest, in which every main claim may cause a random number of delayed claims in an uncertain period of time. Specifically, if both the main claims and delayed claims are subexponential and follow some dependence structure, we drive precise asymptotic formulae for finite-time ruin probabilities, which generalize the results of Lu and Yuan (2022) and Li (2023b) to some extent.

Molecular-Scale Liquid Density Fluctuations and Cavity Thermodynamics.

Equilibrium density fluctuations at the molecular level produce cavities in a liquid and can be analyzed to shed light on the statistics of the number of molecules occupying observation volumes of increasing radius. An information theory approach led to the conclusion that these probabilities should follow a Gaussian distribution. Computer simulations confirmed this prediction across various liquid models if the size of the observation volume is not large. The reversible work required to create a cavity and the chance of finding no molecules in a fixed observation volume are directly correlated. The Gaussian formula for the latter probability is scrutinized to derive the changes in enthalpy and entropy, which arise from the cavity creation. The reversible work of cavity creation has a purely entropic origin as a consequence of the solvent-excluded volume effect produced by the inaccessibility of a region of the configurational space. The consequent structural reorganization leads to a perfect compensation of enthalpy and entropy changes. Such results are coherent with those obtained from Lee in his direct statistical mechanical study.

Non-target liquid chromatography high-resolution mass spectrometry screening to prioritize unregulated micropollutants that persist through domestic wastewater treatment

Efforts to regulate and monitor emerging contaminants are insufficient because new chemicals are continually brought to market, and many are unregulated and potentially harmful. Domestic wastewater treatment plants are not designed to remove micropollutants and are important sources of emerging contaminants in the aquatic environment. In this study, non-target screening, an unbiased method for analyzing compounds without prior information, was used to identify compounds that may be emitted in wastewater treatment plant effluent and should be monitored. Nine wastewater treatment plants using different treatment methods were studied, and a non-target screening data-processing method was used. The frequencies at which the contaminants were detected and contaminant persistence through the treatment processes were considered, and then the contaminants were prioritized. The predicted no-effect concentration of each prioritized contaminant was used to determine whether further analysis and monitoring of the contaminant was necessary. Quantitative analyses of five compounds (amantadine, atenolol, benzotriazole, diphenhydramine, and sulpiride) were performed using reference standards. Probable molecular formulae and structures were proposed for 17 contaminants, and the risks posed by the contaminants were estimated using predicted no-effect concentrations. The results provide valuable insights into how unregulated micropollutants can be identified and prioritized for monitoring in future studies.

Investigation of two-proton decay using modified formation probability

In this study, we investigated two-proton radioactivity using the two-potential approach with a cosh-type potential to calculate the half-lives. The depth parameter V0 = 58.405 MeV and diffuseness a = 0.537 fm in the cosh-type nuclear potential show the lowest standard deviation between the calculated and experimental half-lives. We proposed a linear formula for the formation probability using the linear relationship between log10S2p and A1/3d for the angular momentum state l = 0, 2 and 4. The model achieved the lowest standard deviation (σ = 1.09) using this linear formula compared to previous models and empirical formulas. The proposed formula significantly improved the accuracy of half-life predictions by reducing the standard deviation from 1.73 to 1.09. The predicted half-lives exhibit a hindrance factor in the range of -1.62 to 2.42, which is the lowest compared to earlier theoretical predictions. These results indicate that the proposed linear formation probability formula is suitable for reproducing experimental half-lives. The linear formula for formation probability was generalized for different angular momentum states by conducting least squares fit. We extended the half-life and formation probability predictions to 48 nuclei, and the predicted half-lives are in good agreement with the previous five theoretical models and two empirical formula predictions.

Empirical risk analysis of mining a Proof-of-Work blockchain

AbstractThe process of mining blocks on a blockchain utilizing a Proof-of-Work consensus mechanism carries inherent risks, particularly when the operational expenses associated with mining exceed the rewards earned. Building on previous findings on mining in pools, this paper delves into the question of whether the theoretical formulas for the ruin probability and the expected value of future surplus obtained under particular model assumptions are indeed validated empirically. In particular, we include the presence of transactions fees in the block rewards in our analysis. We also provide algorithms to fit the involved generalized hyperexponential distributions to actual data. Moreover, we perform a sensitivity analysis for different factors of interest, and we quantify the relevance of incorporating temporal dependence and transaction fees in the model.

Large-scale data generation for quantum cascade laser active-region design with automated wavefunction identification

We introduce an automated wavefunction-identification program to generate large-scale datasets for active-region (AR) design of quantum cascade lasers (QCLs) by using machine learning. Conventional QCL design methods rely on expert knowledge and extensive iterative testing; thus, they are inefficient AR design approaches. Our automated approach identifies crucial wavefunctions within a QCL band diagram rapidly and with high accuracy. Key wavefunctions in the optical-transition, injector-, and extractor-regions, which include, upper- and lower-laser levels, injecting levels, extractor levels, and high-energy leakage-path levels, are identified by using a refined k-means clustering algorithm and tailored probability formulas. We find that an accuracy of >95% can be achieved using this identification program. Leveraging our automated wavefunction identification program, we generated approximately 430 000 QCL structures, identified wavefunctions, and computed basic metrics such as energy differences between various levels. A nominally 8 μm-emitting QCL structure, with each stage comprising 24 layers and a fixed applied electric field of 40 kV/cm, which is expected to be close to that corresponding to the laser threshold, was used in the analysis. The compositions of InGaAs wells and AlInAs barriers were lattice-matched to InP, and only layer thicknesses were varied within empirically derived ranges. Using this dataset, we trained neural networks to map QCL structures to energy level differences and tested the performance. The promising results, with a coefficient of determination, R2, values around 90%, validate both the efficacy of our automated program for generating substantial, usable training data and the capability of our network for QCL-AR design.

A Novel Improved Genetic Algorithm for Multi-Period Fractional Programming Portfolio Optimization Model in Fuzzy Environment

The complexity of historical data in financial markets and the uncertainty of the future, as well as the idea that investors always expect the least risk and the greatest return. This study presents a multi-period fractional portfolio model in a fuzzy environment, taking into account the limitations of asset quantity, asset position, transaction cost, and inter-period investment. This is a mixed integer programming NP-hard problem. To overcome the problem, an improved genetic algorithm (IGA) is presented. The IGA contribution mostly involves the following three points: (i) A cardinal constraint processing approach is presented for the cardinal constraint conditions in the model; (ii) Logistic chaotic mapping was implemented to boost the initial population diversity; (iii) An adaptive golden section variation probability formula is developed to strike the right balance between exploration and development. To test the model’s logic and the performance of the proposed algorithm, this study picks stock data from the Shanghai Stock Exchange 50 for simulated investing and examines portfolio strategies under various limitations. In addition, the numerical results of simulated investment are compared and analyzed, and the results show that the established models are in line with the actual market situation and the designed algorithm is effective, and the probability of obtaining the optimal value is more than 37.5% higher than other optimization algorithms.

Transmission characteristics of partially coherent pin-like optical vortex beams in oceanic turbulence

Based on the Rytov approximation theory, the analytical formulae for the mode detection probability and channel capacity of the partially coherent pin-like optical (PCPO) vortex beams propagating in oceanic turbulence are obtained. The effects of light source parameters and oceanic turbulence parameters on the transmission characteristics of the PCPO vortex beams are analyzed in detail by numerical simulations. According to numerical results, a larger spatial coherence length of the partially coherent source endows the beams with a superior channel capacity performance while accompanied by a decrease in transmission robustness. Meanwhile, PCPO vortex beam with greater phase modulation power parameter and longer wavelength is conducive to enhancing the transmission quality through oceanic turbulence. In addition, the channel capacity of the system can be effectively augmented with the increase of the dissipation rate of kinetic energy per unit mass of fluid, the anisotropy factor, the inner scale radius and the decrease of the mean square temperature dissipation rate, the temperature-salinity contribution ratio. The results also indicate that PCPO vortex beam is a better candidate than Gaussian vortex beam for long-distance transmission. This paper provides a theoretical reference for studying an underwater communication link using PCPO vortex beams as the transmission carrier.

Valuing three-asset barrier options and autocallable products via exit probabilities of Brownian bridge

This paper discusses pricing barrier options and autocallable structured products on underlying three assets via exit probabilities of a three-dimensional Brownian bridge. We derive the marginal exit and bivariate co-exit probabilities. Despite the impossibility of finding an analytical trivariate co-exit probability, its calculation is not regarded as a purely numerical problem. After specifying the component to be numerically evaluated, logistic regression with the Monte Carlo method is adopted to predict it. Extensive numerical experiments of calculating the exit probability of the three-dimensional Brownian motion and pricing complex products demonstrate the effectiveness and efficiency of our approach.

Dynamic serviceability and safety reliability analysis of aging PC girder bridges with non-prestressed reinforcement considering concrete shrinkage, creep and stochastic vehicle load flows

Prestressed concrete (PC) bridges’ service performance displays time-dependency due to several factors including concrete shrinkage, creep, resistance degradation, and stochastic vehicle load flows. These non-stationary vehicle load flows, combined with concrete shrinkage, creep, and resistance degradation, result in sequential alterations of internal forces and deformations in simply supported PC girder bridges, ultimately influencing their dynamic reliability. This study proposes a robust methodology that considers the impact of traffic loading, shrinkage, and creep to analyze the dynamic reliability of simply supported PC girder bridges through the development of a load probability model. A stochastic model for truck loads is generated using motion measurement data, which can be used to calculate bridge internal forces and subsequent quantitative analysis of structural deformation due to shrinkage and creep. Subsequently, the formula for the first exceedance probability under resistance degradation is presented and validated through Monte Carlo simulation. Reliability analysis is conducted for both serviceability and load carrying capacity limit states to assess the impact of shrinkage, creep, and increased mean vehicle load on structural reliability. The proposed method offers the theoretical foundation and practical guidance for assessing the structural health status of in-service bridges, thereby providing valuable insights into their long-term performance and maintenance requirements.

Sequentially Quadratic Surrogate Algorithm for Time-dependent Reliability and Reliability Sensitivity Analysis

Time-dependent reliability and reliability sensitivity analysis in presence of random uncertainty is widespread in equipment structures. To this end, this paper establishes a sequentially quadratic surrogate method. Firstly, the global reliability sensitivity analysis (GRS) is transformed into the classification problem of the time-dependent performance function outputs by means of conditional probability formula. Secondly, referring to the strategy of the Meta-IS method, the Kriging model of time-dependent performance function is employed to construct the importance sampling function to generate the importance sampling (IS) samples of failure domain efficiently. Furthermore, the Kriging model is updated in the IS samples set through the single-loop adaptive Kriging method to realize the accurate identification of the failure indicator function of IS samples, as well as simulation of time-dependent failure probability. Finally, utilize the information of the failure samples obtained by the estimation of time-dependent reliability to evaluate GRS. The proposed algorithm has excellent computational efficiency and applicability due to the conversion of the conditional probability formula, which enables the computational consumption of the time-dependent reliability and GRS analysis independent of the dimensions of the inputs, as well as the Meta-IS method, which improves the sampling efficiency and is applicable to the case of complex implicit performance function. The given examples fully verify the conclusions.

About Logically Probable Sentences

The starting point of this paper is the empirically determined ability to reason in natural language by employing probable sentences. A sentence is understood to be logically probable if its schema, expressed as a formula in the language of classical propositional calculus, takes the logical value of truth for the majority of Boolean valuations, i.e., as a logically probable formula. Then, the formal system P is developed to encode the set of these logically probable formulas. Based on natural semantics, a strong completeness theorem for P is proved. Alternative notions of consequence for logically probable sentences are also considered.