Probability and Statistics Review for Finance, Part II
国外优秀的高等数学教材
国外优秀的高等数学教材高等数学是大多数理工科学生必修的一门课程，它涵盖了微积分、线性代数、概率统计等多个重要概念和技巧。
为了提高学生的数学素养和应用能力，选择一本优秀的高等数学教材至关重要。
在国外，有很多备受推崇的高等数学教材，它们以其严谨的理论体系、易于理解的讲解方式和丰富的例题，成为了学生们学习和研究的宝贵资源。
本文将介绍几本国外优秀的高等数学教材，希望能为学生们在学习高等数学时提供参考和借鉴。
一、《Calculus: Early Transcendentals》《Calculus: Early Transcendentals》是由美国数学家詹姆斯·斯图尔特（James Stewart）所著的一本高等数学教材。
这本教材几乎成为了全球许多大学的高等数学教材标准教材，并且荣获了多个数学教育奖项。
其主要特点包括：1. 结构清晰：教材按照章节和节的结构编排，便于学生系统地学习和复习微积分的各个概念。
2. 知识严谨：该教材注重理论证明和逻辑推导，帮助学生深入理解微积分的原理和定理。
3. 真实应用: 《Calculus: Early Transcendentals》在理论讲解之外，还提供了大量真实世界中的应用例题，帮助学生理解微积分在物理、工程等领域的相关应用。
二、《Linear Algebra and Its Applications》《Linear Algebra and Its Applications》由美国数学家大卫·莱(David C. Lay)所著，是一本系统全面讲解线性代数的经典教材。
其主要特点包括：1. 清晰易懂：教材注重讲解线性代数的基本概念、定理和相关技巧，以简明易懂的语言指导学生。
2. 应用广泛：该教材将线性代数与现实生活中的问题相结合，以应用为导向，帮助学生更好地理解并应用线性代数的概念。
3. 丰富例题：《Linear Algebra and Its Applications》提供了大量的例题和习题，旨在让学生通过实战来加深对线性代数知识的理解和掌握。
统计学SCI杂志最新排名
JOURNAL OF THE ROYAL STATISTICAL SOCIETY
43
SERIES C-APPLIED STATISTICS
Total Cites
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Journal Data Filtered By: Selected JCR Year: 2019 Selected Editions: SCIE,SSCI Selected Categories: 'STATISTICS & PROBABILITY' Selected Category Scheme: WoS
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Statistical Analysis and Data Mining
Stochastics and Partial Differential Equations-Analysis
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and Computations
52
PHARMACEUTICAL STATISTICS
53
INSURANCE MATHEMATICS & ECONOMICS
3,379 2,511 1,221
675 3,131 5,273
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985 2,880
伍德里奇计量经济学第六版答案Chapter 1
CHAPTER 1TEACHING NOTESYou have substantial latitude about what to emphasize in Chapter 1. I find it useful to talk about the economics of crime example (Example 1.1) and the wage example (Example 1.2) so that students see, at the outset, that econometrics is linked to economic reasoning, even if the economics is not complicated theory.I like to familiarize students with the important data structures that empirical economists use, focusing primarily on cross-sectional and time series data sets, as these are what I cover in a first-semester course. It is probably a good idea to mention the growing importance of data sets that have both a cross-sectional and time dimension.I spend almost an entire lecture talking about the problems inherent in drawing causal inferences in the social sciences. I do this mostly through the agricultural yield, return to education, and crime examples.These examples also contrast experimental and nonexperimental (observational) data. Students studying business and finance tend to find the term structure of interest rates example more relevant, although the issue there is testing the implication of a simple theory, as opposed to inferring causality. I have found that spending time talking about these examples, in place of a formal review of probability and statistics, is more successful in teaching the students how econometrics can be used. (And, it is more enjoyable for the students and me.)I do not use counterfactual notation as in the modern “treatment effects” literature, but I do discuss causality using counterfactual reasoning. The return to education, perhaps focusing on the return to getting a college degree, is a good example of how counterfactual reasoning is easily incorporated into the discussion of causality.SOLUTIONS TO PROBLEMS1.1 (i) Ideally, we could randomly assign students to classes of different sizes. That is, each student is assigned a different class size without regard to any student characteristics such as ability and family background. For reasons we will see in Chapter 2, we would like substantial variation in class sizes (subject, of course, to ethical considerations and resource constraints).(ii) A negative correlation means that larger class size is associated with lower performance. We might find a negative correlation because larger class size actually hurts performance. However, with observational data, there are other reasons we might find a negative relationship. For example, children from more affluent families might be more likely to attend schools with smaller class sizes, and affluent children generally score better on standardized tests. Another possibility is that, within a school, a principal might assign the better students to smaller classes. Or, some parents might insist their children are in the smaller classes, and these same parents tend to be more involved in thei r children’s education.(iii) Given the potential for confounding factors – some of which are listed in (ii) – finding a negative correlation would not be strong evidence that smaller class sizes actually lead to better performance. Some way of controlling for the confounding factors is needed, and this is the subject of multiple regression analysis.1.2 (i) Here is one way to pose the question: If two firms, say A and B, are identical in all respects except that firm A supplies job training one hour per worker more than firm B, by how much would firm A’s output differ from firm B’s?(ii) Firms are likely to choose job training depending on the characteristics of workers. Some observed characteristics are years of schooling, years in the workforce, and experience in a particular job. Firms might even discriminate based on age, gender, or race. Perhaps firms choose to offer training to more or less able workers, where “ability” might be difficult to quantify but where a manager has some idea about the relative abilities of different employees. Moreover, different kinds of workers might be attracted to firms that offer more job training on average, and this might not be evident to employers.(iii) The amount of capital and technology available to workers would also affect output. So, two firms with exactly the same kinds of employees would generally have different outputs if they use different amounts of capital or technology. The quality of managers would also have an effect.(iv) No, unless the amount of training is randomly assigned. The many factors listed in parts (ii) and (iii) can contribute to finding a positive correlation between output and training even if job training does not improve worker productivity.1.3 It does not make sense to pose the question in terms of causality. Economists would assume that students choose a mix of studying and working (and other activities, such as attending class, leisure, and sleeping) based on rational behavior, such as maximizing utility subject to the constraint that there are only 168 hours in a week. We can then use statistical methods to。
概率论与数理统计英语
概率论与数理统计英语English: Probability theory and mathematical statistics are two branches of mathematics that deal with the concepts and tools used to understand randomness and uncertainty in various phenomena. Probability theory is concerned with quantifying uncertainty and making predictions about the likelihood of certain events occurring, while mathematical statistics uses probability theory to draw conclusions about populations based on sample data. These two fields are closely related and often used together in applications such as insurance, finance, engineering, and social sciences. Probability theory involves concepts such as random variables, probability distributions, and the laws of large numbers, while mathematical statistics covers topics such as estimation, hypothesis testing, and regression analysis. Together, they provide a framework for understanding uncertainty and making informed decisions in the face of incomplete information.中文翻译: 概率论和数理统计是数学的两个分支，涉及用于理解各种现象中的随机性和不确定性的概念和工具。
各专业课程英文翻译
各专业课程英⽂翻译各专业课程英⽂翻译（精⼼整理）⽣物及医学专业课程汉英对照表细胞⽣物学和分⼦⽣物学 Celluar and Molecular Biology 精神病护理学 Psychiatric Nursing⼒学专业⾼等代数与⼏何 Advanced Algebra and Geometry 数学物理⽅法 Methods in Mathematical Physics理论⼒学 Theoretical Mechanics 弹性⼒学 Elasticity⼒学实验 Experiments in Solid Mechanics ⼒学概论Introduction to Mechanics 计算流体⼒学 Computational Fluid Mechanics 粘性流体⼒学Viscous Fluid Flow弹性⼒学变分原理 Variational Principles inElasticity有限元法 Finite Element Method进化⽣物学 Evolutionary Biology ⼝腔外科学 Oral Surgery海洋⽣物学 Marine Biology ⼝腔 / ⽛科科学 Oral/Dental Sciences 微⽣物学 Microbiology⾻科医学 Osteopathic Medicine 分⼦⽣物学 Molecular Biology ⽿科学 Otology医学微⽣物学 Medical Microbiology 理疗学 Physical Therapy ⼝腔⽣物学 Oral Biology ⾜病医学Podiatric Medicine寄⽣物学 Parasutology眼科学 Ophthalmology 植物⽣物学 Plant Physiology 预防医学Preventive Medicine ⼼理⽣物学 Psychobiology 放射学 Radiology放射⽣物学 Radiation Biology 康复咨询学 Rehabilitation Counseling 理论⽣物学 Theoretical Biology 康复护理学Rehabilitation Nursing 野⽣⽣物学 Wildlife Biology 外科护理学 Surgical Nursing 环境⽣物学 Environmental Biology 治疗学Therapeutics 运动⽣物学 Exercise Physiology 畸形学 Teratology有机体⽣物学 Organismal Biology 兽医学 Veterinary Sciences ⽣物统计学 Biometrics ⽛科卫⽣学 Dental Sciences ⽣物物理学 Biophysics ⽛科科学 Dentistry ⽣物⼼理学 Biopsychology ⽪肤学 Dermatology ⽣物统计学 Biostatistics 内分泌学Endocrinology ⽣物⼯艺学 Biotechnology 遗传学 Genetics ⽣物化学 Biological Chemistry解剖学 Anatomy⽣物⼯程学 Biological Engineering ⿇醉学 Anesthesia ⽣物数学 Biomathematics临床科学 Clinical Science应⽤⽣物学 Applied Biology 细胞⽣物学 Cell Biology ⽣物学 Biology⽣物医学科学 Biomedical Science临床⼼理学 Clinical Psychology 医学技术 Medical Technology 医学 Medicine护理⿇醉学 Nurse Anesthesia 数学分析 Mathematical Analysis 常微分⽅程 Ordinary Differential Equation 计算⽅法Numerical Methods 材料⼒学 Mechanics of Materials 流体⼒学Fluid Mechanics 机械制图 Machining Drawing ⽓体⼒学 Gas Dynamics 弹性板理论 Theory of Elastic Plates塑性⼒学Introduction of Plasticity经典⼒学中的数学⽅法 Mathematical Methods of ClassicalMechanics机器⼈动⼒学 Dynamics of Robots ⾃动控制原理 Principles of Automatic Control优化计算与最化控制 Optimization and OptimalControl计算机图形学 Computer Graphics 概率与统计 Probability and Statistics 专业英语 English for Mechanics 振动理论 Theory of Vibration 程序设计⽅法 (C 和 FORTRAN) Programming in C & FORTRAN ⽔动⼒学 Hydrodynamics 计算机图象处理 Image Processing 光测⼒学 Photo Mechanics断裂⼒学 Fracture Mechanics⾼等动⼒学 Advanced Dynamics 摄动⽅法 Perturbation Methods 机械设计与 Auto CADMachinery Designing and AutoCAD 信息显⽰ (可视化 ) Visualization 微机原理 Principles of Personal Computer 复变函数Complex Function企业管理专业微观经济学 Microeconomics管理信息系统 Systems of Management Information 财务管理 Financial Management 战略管理 Strategic Management 国际商务谈判 Negotiation on BusinessAffairs跨国公司专题研究 Special Researchof multinational corporation国际贸易 InternationalTrade 国际营销研究 International Marketing Research 公司组织与管理 Organization and Managementof Corporate信息管理概论 Introduction to InformationManagement信息经济学 Information Economics 企业信息化⼯程管理学原理 Principles of Management 信息政策与法规 Information Policy and Law 管理信息系统 Management Information System 线性代数 Linear Algebra决策分析 Policy Making 离散数学 Discrete Mathematics概率统计 Statistics and ProbabilityTheory ⽣产与运作管理 Production Management 电⼦商务Electronic Commerce 信息系统安全与保密 Information System Security政府信息化⼯程 Government Informationalization ⼴告实务 Practice of Advertisement 操作系统 Operating System 信息科学基础Foundations of InformationScience 经济信息管理 Economic Information Management 专业英语 Specialty English 微机基础Principles of Microcomputers ⽂献计量学 Bibliometrics电⼦出版技术 Electronic Publishing ⼴告概论 Introduction to Advertisement管理学 Principles of Management 宏观经济学 Macroeconomics 产业经济学 Industrial Economics 项⽬评估 Projects Appraisal 管理沟通Management Negotiation战略管理 Strategic Management ⽣产管理研究 Operation Management 企业伦理 Enterprise Ethics 运筹学 Operational Research 信息管理专业⾼等数学 Higher Mathematics数据库系统 Database 组织⾏为学 Organizational Behavior⼈⼒资源管理 Human Resource Management信息存储与检索信息服务与Information Retrieval andStorage information Service and UserStudy社会实践 Practical Work信息分析与决策 Information Analysis andPolicy Making Enterprise Informationalization 信息组织 Information Organization 计算机⽹络 Computer Networks 多媒体技术 Multimedia信息环境论 Information Environments 传播学原理 Principles of CommunicationTheory 知识产权法学 Law of Intelligence Property 组织⾏为学 Studies of Organization 货币银⾏学专业货币银⾏学 Money and Banking 宏观经济学 Macroeconomics 策略管理 Strategic Management 银⾏会计 Bank Accounting 运筹学 Operational Research 财务管理Financial Management 租赁与信托 Hiring and Affiancing 商业银⾏实务 Practice of Business Bank项⽬评估 Projects Appraisal ⾦融市场学⼈⼒资源管理 Human ResourceManagement 财务报告分析 A nalysis of Financial Statement财务案例分析 Case Analysis of FinancialManagement物理专业热学 Thermodynamics ⼒学 Mechanics光学 Optics电磁学 Electromagnetism计算概论 Computing Generality 普通物理实验 General Physics Laboratory固体磁性及应⽤基础 Magnetism of the Solid Stateand its Application衍射物理(固体结构分析)Diffraction Physics (Structureof Solid Analysis)科研实⽤软件 Utility Software for ScientificResearch 计算机模拟⽅法 Computer SimulationMethods 激光原理、技术与应⽤ The Principle, Techniqueand Application of Laser材料物理 Materials Physics 近代光学和光电⼦学 Modern Optics and Optoelectronics现代固体物理 Modern Solid State Physics粒⼦物理 Particle Physics 物理宇宙学基础 Elements of Cosmology Physics 固体物理 Solid State Physics 原⼦物理 Atomic Physics量⼦⼒学 Quantum Mechanics 理论⼒学 Theoretical Mechanics电动⼒学 Electrodynamics普通物理综合实验 Synthetical Experiments ofGeneral Physics市场营销学专业营销管理 Marketing Management 公共关系 Public Relationship 国际贸易 International Trade 消费者⾏为 Consumer Behavior 管理信息系统 Systems of Management Information 营销调研Marketing Research 推销学 Sales Strategies 国际⾦融 International Finance 营销预测与规划 Marketing Forecasting andPlanning 销售渠道管理 ales Channels Management 管理学 Principles of Management 国际市场营销 International Marketing 商业谈判 Business Negotiation ⼴告管理 Advertising Management 营销案例分析 Case Studies of Marketing 国际贸易实务 Practice of InternationalTrade 服务业营销 Service Industry Marketing企业伦理 Enterprise Ethics 新产品开发 New Products Development管理信息系统 System of Management Information 运筹学 Operational Research 保险学 Insurance管理会计 Managerial Accounting 国际贸易 International Trade国际⾦融 International Finance证券投资学 Security Analysis and Investment 国际结算 International BalanceFinancial Marketing证券投资学Security Analysis and Investment 财务学专业财务报告分析Analysis of Financial Statement 国际⾦融International Finance保险学Insurance 财务案例分析Case Analysis of FinanceManagement 国际财务管理International Financial Management 资产评估Assets Appraisal项⽬评估Projects Appraisal 财务管理Financial Management 运筹学Operational Research 管理会计Managerial Accounting 管理学Principles of Management 统计学Principles of Statistics宏观经济学Macroeconomics管理信息系统Systems of Management Information 策略管理Strategic Management 微观经济学Microeconomics 微积分Calculus会计专业会计学Accounting Principles 管理会计Managerial Accounting 会计信息系统Accounting Information Systems 财务管理Financial Management 财务报告分析Analysis of Financial Statement 国际会计International Accounting成本会计Cost Accounting 审计学Auditing Principles 投资学Investment Principles 货币银⾏学Money and Banking国际⾦融International Finance 统计学Principle of Stat财税法规与税务会计Laws and Regulations of Financeand Taxes预算会计Budget Accounting 会计研究⽅法Accounting Research Methods 内部审计与政府审计Internal Auditing and GovernmentAuditing会计审计实务Accounting and Auditing Practice 经济计量学Economic Metrology会计职业道德与责任Accounting Ethics and Responsibilities国际会计专题International AccountingSpecial Subject 微观经济学Microeconomics 货币银⾏学Money and Banking。
FRM 金融风险管理师考试大纲
Quantitative Analysis Readings:
1. Linda Allen, Jacob Boudoukh, Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value At Risk Approach (Oxford: Blackwell Publishing, 2004). Chapter 2 – Quantifying Volatility in VaR Models th John Hull, Options, Futures, and Other Derivatives, 6 ed. (New York: Prentice Hall, 2006). Chapter 19 – Estimating volatilities and correlations rd Philippe Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3 ed. (New York: McGraw-Hill, 2007). Chapter 9 – Forecasting risk and correlations Chapter 12 – Monte Carlo Methods Lampros Kalyvas and Ioannis Akkizidis, Integrated Market, Credit and Operational Risk: A Complete Guide for Bankers and Risk Professionals (London: Risk Books, 2006). Chapter 4 – Extreme Value Theory and in Risk Management Murray R. Spiegel, John Schiller, and R. Alu Srinivasan, Probability and Statistics, Schaum’s Outlines, 2nd ed. (New York: McGraw-Hill, 2000). Chapter 1 – Basic Probability Chapter 2 – Random Variables and Probability Distributions Chapter 3 – Mathematical Expectation Chapter 4 – Special Probability Distributions Chapter 5 – Sampling Theory Chapter 6 – Estimation Theory Chapter 7 – Tests of Hypotheses and Significance Chapter 8 – Curve Fitting, Regression, and Correlation NOTE: Candidates should not memorize formulas of distributions but should understand when it is appropriate to use a particular type of distribution.
02 统计学书单
1. Advanced Calculus with Applications in Statistics2.A History of Probability and Statistics and Their Applications before 17503.Markov Decision Processes: Discrete Stochastic Dynamic Programming4.Probability and Statistical Inference5.Continuous Univariate Distributions, V ol. 16.Continuous Univariate Distributions, Vol. 27.The Theory of Measures and Integration8.Robust Statistics: Theory and Methods9.Finite Mixture Models10.Generalized, Linear, and Mixed Models11.Statistics of Extremes: Theory and Applications12.Modes of Parametric Statistical Inference13.Univariate Discrete Distributions14.Contemporary Bayesian Econometrics and Statistics15.Approximation Theorems of Mathematical Statistics16.Image Processing and Jump Regression Analysis17.Operational Risk : Modeling Analytics18.Design and Analysis of Experiments, Introduction to Experimental Design19.Introductory Biostatistics for the Health Sciences: Modern ApplicationsIncluding Bootstrap20.Linear Models in Statistics21.Statistics for Research22.Applied Logistic Regression23.Operational Subjective Statistical Methods: A Mathematical, Philosophical,and Historical Introduction24.Probability and Measure, 2nd Edition25.Theory of Preliminary Test and Stein-Type Estimation with Applications26.The EM Algorithm and Extensions27.The Theory of Response-Adaptive Randomization in Clinical Trials28.Models for Probability and Statistical Inference: Theory and Applications29.Applied Life Data Analysis30.Structural Equation Modelling: A Bayesian Approach31.Bootstrap Methods: A Guide for Practitioners and Researchers32.Nonparametric Analysis of Univariate Heavy-Tailed Data: Research andPractice33.Applied Linear Regression, 3rd edition34.Theory of Probability: A Critical Introductory Treatment35.Financial Derivatives in Theory and Practice36.Quantitative Methods in Population Health: Extensions of Ordinary Regression37.Statistical Methods for Survival Data Analysis38.Applied Bayesian Modelling39.Spatial Statistics, 2004-0840.Approximate Dynamic Programming: Solving the Curses of Dimensionality41.Variance Components42.Time Series: Applications to Finance43.Generalized Least Squares44.Statistical Analysis With Missing Data45.Long-Memory Time Series: Theory and Methods46.Statistical Models and Methods for Lifetime Data47.Uncertainty Analysis with High Dimensional Dependence Modelling48.Simulation and the Monte Carlo Method49.A Matrix Handbook for Statisticians50.Meta Analysis: A Guide to Calibrating and Combining Statistical Evidence51.Precedence-Type Tests and Applications52.Statistical Meta-Analysis with Applications53.Management of Data in Clinical Trials54.Periodically Correlated Random Sequences: Spectral Theory and Practice55.Design and Analysis of Experiments, Advanced Experimental Design56.Methods and Applications of Linear Models : Regression and the Analysis ofVariancebinatorial Methods in Discrete Distributions58.Nonparametric Regression Methods for Longitudinal Data Analysis:Mixed-Effects Modeling Approaches59.Response Surfaces, Mixtures, and Ridge Analyses60.Variations on Split Plot and Split Block Experiment Designs61.Recent Advances in Quantitative Methods in Cancerand Human Health Risk Assessment62.The Construction of Optimal Stated Choice Experiments: Theory and Methods63.Nonparametric Density Estimation: The L1 View64.Applied MANOV A and Discriminant Analysis65.Survey Errors and Survey Costs66.Statistical Advances in the Biomedical Sciences: Clinical Trials,Epidemiology, Survival Analysis, and Bioinformaticstent Curve Models: A Structural Equation Perspective68.Regression Diagnostics: Identifying Influential Data and Sources ofCollinearity69.Reliability and Risk: A Bayesian Perspective70.Environmental Statistics71.Bayes Linear Statistics, Theory & Methods72.Introductory Stochastic Analysis for Finance and Insurance73.Bayesian Models for Categorical Data74.Bayesian Statistical Modelling75.Weibull Models76.Analysis of Financial Time Series77.Linear Model Theory: Univariate, Multivariate, and Mixed Models78.An Introduction to Categorical Data Analysis79.Bayesian Statistics and Marketing80.Statistical Shape Analysis81.Nonparametric Statistics with Applications to Science and Engineering82.Longitudinal Data Analysis83.Regression Models for Time Series Analysis84.Introduction to Nonparametric Regression85.Statistical Modeling by Wavelets86.Case Studies in Reliability and Maintenance87.The Geometry of Random Fields88.Biostatistics : A Methodology For the Health Sciences89.Planning, Construction, and Statistical Analysis of Comparative Experiments90.Statistical Size Distributions in Economics and Actuarial Sciences91.Modern Experimental Designparative Statistical Inference93.Methods of Multivariate Analysis94.Robust Statistics95.Order Statistics96.Fourier Analysis of Time Series : An Introduction97.Spatial Statistics 1981-0498.Clinical Trials : A Methodologic Perspective99.Numerical Issues in Statistical Computing for the Social Scientist100.Nonlinear Regression101.Preparing for the Worst : Incorporating Downside Risk in Stock MarketInvestments 102.Nonlinear Regression Analysis and Its Applications103.Mixed Models : Theory and Applications104.Discrete Distributions : Applications in the Health Sciences105.Design and Analysis of Clinical Trials : Concepts and Methodologies106.Applied Bayesian Modeling and Causal Inference from Incomplete-DataPerspectives 107.Flowgraph Models for Multistate Time-?–to-Event Data108.Randomization in Clinical Trials: Theory and Practice109.Regression With Social Data: Modeling Continuous and Limited ResponseVariables 110.Regression Analysis by Example111.Categorical Data Analysis112.Statistical Methods for Reliability Data113.Elements of Stochastic Processes Wit114.Random Graphs for Statistical Pattern Recognition115.Construction and Assessment of Classification Rules116.The Statistical Analysis of Failure Time Data117.Statistical Analysis of Finite Mixture Distributions118.Modern Applied U-Statistics119.Applied Multiway Data Analysis。
Probability and Stochastic Processes
Probability and Stochastic Processes Probability and stochastic processes are fundamental concepts in the field of mathematics and statistics. They play a crucial role in various real-world applications, ranging from finance and economics to engineering and biology. Understanding these concepts is essential for making informed decisions in uncertain situations and predicting future outcomes. Probability is thelikelihood of a particular event occurring, expressed as a number between 0 and 1. It provides a quantitative measure of uncertainty and helps us make sense of random phenomena. Stochastic processes, on the other hand, are collections of random variables that evolve over time. They allow us to model complex systemsthat exhibit random behavior, such as stock prices or weather patterns. One of the key principles in probability theory is the law of large numbers, which states that as the number of trials increases, the observed frequency of an event converges to its theoretical probability. This concept is essential for understanding the behavior of random variables in the long run and forming the basis for statistical inference. Another important concept in probability theory is conditional probability, which quantifies the likelihood of an event given that another event has already occurred. It is crucial for making predictions in situations where certain information is known, such as medical diagnosis or weather forecasting. Conditional probability also plays a key role in Bayesian statistics, where prior knowledge is used to update beliefs about the likelihood of different outcomes. Stochastic processes come in various forms, such as Markov chains, random walks, and Brownian motion. These processes are used to model a wide range of phenomena, from the movement of particles in a fluid to the behavior of financial markets. Markov chains, in particular, are widely used inapplications such as speech recognition, text processing, and image analysis. In finance, stochastic processes are used to model the behavior of asset prices and interest rates. These models help investors and financial institutions make informed decisions about risk management, portfolio optimization, and derivative pricing. By simulating different scenarios using stochastic processes, analysts can assess the potential impact of market fluctuations and develop strategies to mitigate risks. In conclusion, probability and stochastic processes are powerfultools for analyzing uncertainty and making informed decisions in a wide range of fields. By understanding the principles of probability theory and mastering the techniques of stochastic modeling, we can gain valuable insights into the behavior of complex systems and improve our ability to predict future outcomes. Whether we are dealing with stock prices, weather patterns, or biological processes, probability and stochastic processes provide us with the tools we need to navigate the uncertainties of the world around us.。
Grms 加速度平方根值
Calulating G rms(Root-Mean-Square Acceleration)It is very easy to describe the G rms (root-mean-square acceleration, sometimes written as GRMS or Grms or grms or g rms) value as just the square root of the area under the ASD vs. frequency curve, which it is. But to physically interpret this value we need to look at G rms a different way. The easiest way to think of the G rms is to first look at the mean square acceleration.其实是很容易来描述这个G rms（根均方加速）作为只是平方根值下的房间隔缺损与频率曲线。
但是，从整体上解释这个值，我们需要以不同的方式认识G rms。
最简单的方法是思考G rms加速度均方。
Mean-square acceleration is the average of the square of the acceleration over time. That is, if you were to look at a time history of an accelerometer trace and were to square this time history and then determine the average value for this squared acceleration over the length of the time history, that would be the mean square acceleration. Using the mean square value keeps everything positive.均方加速是在一段时间内加速平方的平均值。
Ⅱ型删失下Lindley分布的参数估计
Ⅱ型删失下Lindley分布的参数估计龙兵【摘要】在Ⅱ型删失数据下,讨论了Lindley分布参数的最大似然估计.给出了参数的区间估计和逆矩估计,运用随机模拟的方法对参数进行了统计分析.通过一个例子求出了在不同Ⅱ型删失样本下参数的两种点估计及区间估计,并进行了比较.%Maximum likelihood estimation of the parameter from the Lindley distribution was dis-cussed under Ⅱ censored data. The interval estimation and inverse moment estimation were given, and the random simulation method was used to analyze the parameter. An example was provided to find out the two kinds of point estimation and interval estimation about the parameter under differ ent typeⅡcen-sored parison between the two point estimations was discussed.【期刊名称】《湖南师范大学自然科学学报》【年(卷),期】2017(040)006【总页数】5页(P71-75)【关键词】Lindley分布;χ2分布;区间估计;逆矩估计;最大似然估计【作者】龙兵【作者单位】荆楚理工学院数理学院,中国荆门 448000【正文语种】中文【中图分类】O212.2Lindley distribution was proposed by Lindley in the literature[1-2] in 1958, which is an important distribution in reliability study. For some life data, the use of Lindley distribution model to fit the effect will be better. At present, there are a lot of statistical scholars who have discussed its properties, and obtained a lot of research results. The properties and applications of compound Lindley distribution were studied in the literatures [3-5]. The empirical Bayes one-sided test was discussed in the literature [6] in the case of independent identically distributed sample. The EB test function was constructed by using the recursive kernel estimation of density function. The optimality of the test function was proved, and the convergence rate was obtained. The empirical Bayes test function of the parameter from Lindley distribution was discussed based on NA random sample sequence in the literature [7]. The interval estimation and hypothesis testing of the parameter in Lindley distribution were studied in the full sample in the literature [8]. The failure model of incomplete data with Lindley distribution was discussed in the literature[9],and maximum likelihood estimation method was used to obtain point estimation and asymptotic confidence interval estimation of the model. An example was given to show that the Lindley distribution have better adaptability compared with exponential distribution and Weibull distribution.In this paper, we will discuss the maximum likelihood estimation, interval estimation and inverse moment estimation of the parameter from Lindley distribution in Ⅱ censored samples, the parameter will be estimated by the method of stochastic simulation, and the average deviation will becalculated. Finally, an example will be given to illustrate the feasibility of the proposed method.The probability density function of Lindley distribution isf(x)=(1+x)e-θx， x>0.Its distribution function isF(x)=1-(1+x)e-θx ，x>0,with parameter θ>0.The data in life test for some data will usually be censored, leading to incomplete data, namely, censored way are type I censored, type Ⅱ censored and so on.It is assumed that there are n products that are independent of each other and are subject to Lindley distribution, with type Ⅱ censored test. When observed with m failure samples, the remaining n-m samples have been withdrawn from the test. The failure time of the m samples have been observed to meet X(1)≤X(2)≤…≤X(m). For the sake of convenience, we will omit the parenthesis of the subscript numbers. The Xi represents the minimum i observation value, which is the full sample case when m=n. According to the above test, the likelihood function is).Substituting (1) and (2) to (3) gives).Log likelihood function islnL(θ)=lnA+2mlnθ-mln(θ+1)+(n-m)ln(1+xm)-θ[xi+(n-m)xm],].Set up h1(θ)=+，h2(θ)=+[xi+(n-m)xm].Obviously h1(θ)>0, h2(θ)>0. When θ→0+,h1(θ)>h2(θ)So, h1(θ) is strictly monotone decreasing lower convex function on (0，+∞) andTherefore, h2(θ) is also strictly monotone decreasing lower convex function on (0，+∞),and h1(θ)<h2(θ)(θ→+∞), indicating that].There is a unique solution on (0，+∞), which is maximum likelihood estimation of the parameter θ, denoted as .According to the distribution function of Lindley distribution, we can get the following lemma.Lemma 1 Let random variable X follow Lindley distribution with parameter θ and let Y=θX-ln(1+X),then Y follows standard exponential distribution. Proof：Let FX(x),FY(y) denote the distribution function of random variable X and Y, respectively, thenFX(x)=1-(1+x)e-θx ，x>0.Therefore 1-(1+X)e-θX～U(0,1)FY(y)=P(Y≤y)=P{θX-ln(1+)≤y}=P{(1+X)e-θX≥e-y}=P{1-(1+X)e-θX≤1-e-y}=1-e-y，y>0.The above expression is just the distribution function of standard exponential distribution, proving Lemma 1.According to the above test, let X(1),X(2),…，X(m) be a type Ⅱ censored sample from Lindley distribution (2) with a capacity of n. One can obtain from Lemma 1 thatθX(1)-ln(1+X(1))，θX(2)-ln(1+X(2))，…，θX(m)-ln(1+X(m))is a type Ⅱ censored sample from standard exponentialy distribution (2) with a capacity of n.Set upW1=n[θX(1)-ln(1+X(1))],W2=(n-1){[θX(2)-ln(1+X(2))]-[θX(1)-ln(1+X(1))]},……Wm=(n-m+1){[θX(m)-ln(1+X(m))]-[θX(m-1)-ln(1+X(m-1))]}.According to literature [10], W1,W2,…，Wm are independent and identically distributed and follow standard exponential distribution. Lemma 2 Let W1,W2,…，Wm be independent and identically distributed and follow standard exponential distribution and set up Sm=2Wj,j=1,2,…，m，then Sm follows Chi-square distribution with 2m degree of freedom. Proof: Because Wj,j=1,2,…，m follows standard exponential distribution, its characteristic function isThe above formula is the characteristic function of the Chi-square variable with 2m degree of freedom. Lemma 2 is proved by the unique decision of characteristic function and the distribution of random variable.It can be proved that Wj,j=1,2,…，m is a monotonically increasing function of θ, so Sm is a monotonically increasing function of θ.Theorem 1 Let X(1),X(2),…，X(m) be a type Ⅱ censored sample from Lindley distribution (2) with a capacity of n, for any 0<α<1, under the confidence level of 1-α,the confidence interval of θ iswhere (·) is an inverse function of Sm, and are respectively and 1-quantiles on Chi-square distribution with 2m degree of freedom.Proof According to Lemma 1, Sm～χ2(2m), and Sm is a monotonically increasing function of θ.Therefore,P(((2m))≤θ≤((2m)))=P((2m)≤Sm≤(2m))=1-α.Because W1,W2,…，Wm are independent and identically distributed and follow standard exponential distribution, it can be used as a quasi sample. Thus, inverse moment estimation of the parameter θ is determined by the following expression(2m).Set θ=1. A simple random sample is randomly generated with a capacity of n from Lindley distribution (2). Different m values are used to obtain type Ⅱ censored samples. Maximum likelihood estimation and inverse moment estimation of the parameter θ can be obtained by expressions (4) and (5). Repeat the above process 1 000 times to obtain the mean and relative deviation of the parameter estimations. The simulation results are shown in Table 1.The simulation results show that the deviation of inverse moment estimation is smaller than maximum likelihood estimation in small sample case, with the increase of sample size, the deviation of inverse moment estimation is larger than maximum likelihood estimation. In general, the mean of inverse moment estimation is larger than maximum likelihood estimation, therefore, the inverse moment estimation is more effective in small sample situations.Set θ=1, a simple random sample is generated with a capacity of 20 fromLindley distribution (2) as follows 0.111 8，0.291 5, 0.327 9, 0.394 8， 0.531 5, 0.628 4, 0.779 1, 0.848 4, 0.939 5, 1.332 3, 1.475 2, 1.719 0, 1.862 5, 1.900 4, 2.061 6, 2.098 8, 3.130 7, 3.303 7, 4.295 6, 6.593 2. Different m values are used to obtain different type Ⅱ censored samples, and carried on the statistical analysis. The results are listed in Table 2.We can see from Table 2 that when sample size n is fixed, the interval length increases with the decrease of m. Inverse moment estimation is closer to the true value than maximum likelihood estimation.[1] LINDLEY D. Introduction to probability and statistics from a Bayesian viewpoint, part II: inference[M]. Cambridge: Cambridge University Press, 1965.[2] LINDLEY D. Fiducial distributions and Bayes’ theorem[J].J Royal Stat Soc, 1958,20(1):102-107.[3] GHITANY M E, ATIEH B, NADARAJAH S. Lindley distribution and its application[J]. Math Comput Simula, 2008,78(4):493-506.[4] GHITANY M E, ALMDK, NADARAJAH S. Zero-truncated Poisson-Lindley distribution and its application[J]. Math Comput Simula, 2008,79(3):279-287.[5] ZAMANI H, ISMAIL N. Negative binomial-Lindley distribution and its application[J].J Math Stat, 2010,6(1):4-9.[6] 杜伟娟，彭家龙，李体政.Lindley分布参数的经验Bayes检验的收敛速度[J].统计与决策，2012,21:23-26.[7] 范梓淼，周菊玲. NA 样本下Lindley 分布参数的经验Bayes 检验[J].贵州大学学报(自然科学版)，2016,34(2):68-70.[8] 龙兵. Lindley分布中参数的区间估计和假设检验[J].广西民族大学学报(自然科学版)，2014,20(1):59-62.[9] 黄文平，周经伦，宁菊红,等.基于竞争失效数据的Lindley分布参数估计[J].系统工程与电子技术，2016,38(2):464-469.[10] LAWLESS J F. Statistical models and methods for lifetime data[M].New York: Wiley, 2003.【相关文献】DOI:10.7612/j.issn.1000-2537.2017.06.012。