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Kolmogorov II

Students at all levels of study in the theory of probability and in the theory of statistics will find in this book a broad and deep cross-section of problems (and their solutions) ranging from the simplest combinatorial probability prob­ lems in finite sample spaces through information theory, limit theorems and the use of moments. The introductions to the sections in each chapter establish the basic formulas and notation and give a general sketch of that part of the theory that is to be covered by the problems to follow. Preceding each group of problems, there are typical examples and their solutions carried out in great detail. Each of these is keyed to the problems themselves so that a student seeking guidance in the solution of a problem can, by checking through the examples, discover the appropriate technique required for the solution.

Students at all levels of study in the theory of probability and in the theory of statistics will find in this book a broad and deep cross-section of problems (and their solutions) ranging from the simplest combinatorial probability prob­ lems in finite sample spaces through information theory, limit theorems and the use of moments. The introductions to the sections in each chapter establish the basic formulas and notation and give a general sketch of that part of the theory that is to be covered by the problems to follow. Preceding each group of problems, there are typical examples and their solutions carried out in great detail. Each of these is keyed to the problems themselves so that a student seeking guidance in the solution of a problem can, by checking through the examples, discover the appropriate technique required for the solution.

 The contents of the textbook is composed of the lectures in probability theory and mathematical statistics for students of the Faculty of Computational Mathematics and Cybernetics of Moscow University. Probability theory and mathematical statistics are studied for three terms (fifth, sixth and seventh) once a week, accompanied by seminar classes (in the fifth term, it is one class a week, while in the sixth and seventh, one class every other week).   The lectures are, therefore, broken term-wise into three parts, viz., Probability, Stochastic Processes and Statistics. Two hundred problems given at seminar classes are distributed so that Part 1 contains one hundred of them, with Parts 2 and 3 including fifty problems each (i.e., in proportion to the frequency of seminars).   In Part 1 Probability, the accent is on the following.   1. In each example and each problem, a probability space is supposed to be chosen first. In some cases, several probability spaces may be selected for the same problem. We emphasize that probability theory starts only with choosing a probability space, and it is the probability space that is the subject matter of probability theory.   2. At the initial stage of the study of probability theory, combinatorial problems are not cultivated, since the cult of such problems (cuts the ground on which probability theory is constructed from under the student’s feet. Besides, this creates an impression of certain primitiveness and artificiality.   3. The discrete probability space is investigated; we give many statements often formulated only for a general probability space, but, generally, with proofs not altering (e.g., Khinchine law of large numbers).   In Part 2 Stochastic Processes, the choice of material non-traditional. We discuss continuous-time Markov processes with countable state space, renewal and regenerative processes, for it is random processes that most students will subsequently apply. Note, in addition, that the statements and proofs of ergodic theorems for Markov chains do not depend on (either discrete or continuous) time. Little attention is paid to the central limit theorems of probability theory (simplest central limit theorem is given as an example only in connection with the Prokhorov theorem applications).   In Part 3 Statistics, the basic concepts of mathematical statistics are first defined (statistical structure, negligibility, sufficient statistic, complete statistic, free statistic). Then only the theory of point estimation is given in detail (efficient estimators, MLE's, LSM-estimators), and also that of interval estimation (of maximum likelihood on the basis of likelihood ratio principle). The theory of hypothesis testing is recounted quite briefly (students who specialize in mathematical statistics then make up for this gap).  Several proofs and treatment of the material may seem unhabitual to the specialist; however, they are assimilated easier by the beginners. 

 The contents of the textbook is composed of the lectures in probability theory and mathematical statistics for students of the Faculty of Computational Mathematics and Cybernetics of Moscow University. Probability theory and mathematical statistics are studied for three terms (fifth, sixth and seventh) once a week, accompanied by seminar classes (in the fifth term, it is one class a week, while in the sixth and seventh, one class every other week).   The lectures are, therefore, broken term-wise into three parts, viz., Probability, Stochastic Processes and Statistics. Two hundred problems given at seminar classes are distributed so that Part 1 contains one hundred of them, with Parts 2 and 3 including fifty problems each (i.e., in proportion to the frequency of seminars).   In Part 1 Probability, the accent is on the following.   1. In each example and each problem, a probability space is supposed to be chosen first. In some cases, several probability spaces may be selected for the same problem. We emphasize that probability theory starts only with choosing a probability space, and it is the probability space that is the subject matter of probability theory.   2. At the initial stage of the study of probability theory, combinatorial problems are not cultivated, since the cult of such problems (cuts the ground on which probability theory is constructed from under the student’s feet. Besides, this creates an impression of certain primitiveness and artificiality.   3. The discrete probability space is investigated; we give many statements often formulated only for a general probability space, but, generally, with proofs not altering (e.g., Khinchine law of large numbers).   In Part 2 Stochastic Processes, the choice of material non-traditional. We discuss continuous-time Markov processes with countable state space, renewal and regenerative processes, for it is random processes that most students will subsequently apply. Note, in addition, that the statements and proofs of ergodic theorems for Markov chains do not depend on (either discrete or continuous) time. Little attention is paid to the central limit theorems of probability theory (simplest central limit theorem is given as an example only in connection with the Prokhorov theorem applications).   In Part 3 Statistics, the basic concepts of mathematical statistics are first defined (statistical structure, negligibility, sufficient statistic, complete statistic, free statistic). Then only the theory of point estimation is given in detail (efficient estimators, MLE's, LSM-estimators), and also that of interval estimation (of maximum likelihood on the basis of likelihood ratio principle). The theory of hypothesis testing is recounted quite briefly (students who specialize in mathematical statistics then make up for this gap).  Several proofs and treatment of the material may seem unhabitual to the specialist; however, they are assimilated easier by the beginners. 

Lectures in probability theory and mathematical statistics for students of the Faculty of Computational Mathematics and Cybernetics of Moscow University. Revised from the 1983 russian edition

Lectures in probability theory and mathematical statistics for students of the Faculty of Computational Mathematics and Cybernetics of Moscow University. Revised from the 1983 russian edition

Lectures in probability theory and mathematical statistics for students of the Faculty of Computational Mathematics and Cybernetics of Moscow University. Revised from the 1983 russian edition

Lectures in probability theory and mathematical statistics for students of the Faculty of Computational Mathematics and Cybernetics of Moscow University. Revised from the 1983 russian edition

Lectures in probability theory and mathematical statistics for students of the Faculty of Computational Mathematics and Cybernetics of Moscow University. Revised from the 1983 russian edition

677 p. 24 cm

677 p. 24 cm

677 p. 24 cm

677 p. 24 cm

677 p. 24 cm

677 p. 24 cm

677 p. 24 cm

This book is based on a three-term course of lectures that the author delivered at the Faculty of Computational Mathematics and Cybernetics of Moscow University. It is divided into three parts: Probability, Stochastic Processes and Statistics. The text includes approximately 200 problems and illustrative examples and is designed for university and college students, as well as anyone studying or using probability theory. Prof. G.P. Klimov, D.Sc. (Phys.-Math.), is a lecturer in mathematical statistics at Moscow University. His scientific interests lie within the field of random processes, queue theory, invariant statistical processes, and invariant statistical inference. He is the author of more than 60 papers and several books translated into German, Polish, and Bulgarian. Prof. G.P. Klimov has also delivered courses of lectures at the Sorbonne, Universities of Liège, Berlin, and the Stefan Banach International Mathematical Centre in Warsaw. Translated from the Russian by Oleg Efimov All credits to IA user  Quantum AI Okay, here’s the table of contents formatted for copy-pasteability into an RTF document. It includes bullet points to clearly delineate each section and subsection, making it easy to use within a document editor.     Table of Contents              Part 1: Probability Theory              Introduction          Chapter 1: Discrete Probability Space              Event and Probability              Probability Space              Independence of Events              Conditional Probability              Probability of Union of Events              Composite Probability Formula              Bayes Formula              Examples of Distributions              Random Variable, Mathematical Expectation              Independence of Random Variables              Additive and Multiplicative Properties of Mathematical Expectation              Moments              Variance Additivity for Independent RV’s              Generating Function              Binomial Distribution              Poisson Distribution              Negative Binomial Distribution              Approximation of Binomial by Poisson Distribution              Indicator of Event              Markov Inequality              Law of Large Numbers              Central Limit Theorem          Chapter 2: Probability Space              Definition of Probability Space              Construction of Probability Space              Complete Probability Space              Continuous Probability Space              Definition of Random Variable              Operations over RV’s              Equivalence and Almost Certain Convergence of RV's              Independence              Mathematical Expectation, Distribution Function, Distribution Density          Chapter 3: Subjective Probability (Supplement)              Semi-Markov Renewal Process          Part 2: Stochastic Processes              Introduction: Definition of Random Process, Kolmogorov Theorem on Construction of Processes from Partial Distributions          Chapter 3: Discrete- and Continuous-Time Markov Chains              Definition of Discrete- and Continuous-Time Markov Chains              Examples of Homogeneous Markov Chains              Limiting Distribution of Markov Chains          Chapter 4: Renewal Processes              Definition of Renewal Process              Elementary Renewal Theorem              Blackwell Theorem              Key Renewal Theorem          Chapter 5: Regenerative Processes              Definition of Regenerative Process              Limit Theorem for Regenerative Process              Law of Large Numbers for Regenerative Processes          Chapter 6: Elements of Random Analysis              Types of RV Convergence              Equivalent Definitions of RV Convergence and Basic Properties              Prokhorov Theorem and Its Applications              Stochastic Continuity              Stochastic Integrals of Non-Random Functions              Orthogonal Measure and Integral with Values in Hilbert Space              Differentiation and Integration of Mathematical Expectation of Processes          Chapter 7: Gaussian Processes              Definition of Gaussian Process              Wiener Process              Non-Homogeneous Wiener Process          Part 3: Statistics              Introduction          Chapter 9: Mathematical Statistics: Basic Concepts              Statistical Structure              Statistic              Sufficient Statistic              MVUE (Most Variable Unbiased Estimator)              Complete and Sufficient Statistic              Minimum Risk Criterion          Chapter 10: Point Estimation – Cramér-Rao Theory              MVUE (Most Variable Unbiased Estimator)              MVUE Uniqueness              Cramér-Rao Inequality              Efficient Estimators              Multidimensional Analogue of Cramér-Rao Inequality          Chapter 11: Point Estimation – Maximum Likelihood Method              Maximum Likelihood Estimator              Consistent Estimators              MLE Consistency              MLE Asymptotic Normality          Chapter 12: Point Estimation – Least Squares Method              Examples              Normal Equation              Gauss-Markov Theorem              Linear Regression Model              Normal Error Distribution          Chapter 13: Interval Estimation              Definition of Confidence Intervals              Likelihood Ratio Principle              Technique of Likelihood Ratio Principle Application          Chapter 14: Linear Model – Multidimensional Analysis              Introduction              Assumptions for Space of Parameters (6 = (n, a))              MLE Property              Estimating Function              Interval MLE for 6 = (n, a)              Independence of nm and a              Estimating pp and a Separately          Chapter 15: Statistical Hypothesis Testing              Hypothesis and Test              Power Function, Test Ordering              Fundamental Neyman-Pearson Lemma          Chapter 16: Sequential Analysis              Problem Posing              Probability Ratio Sequential Test for s Hypotheses              Average Sample Size Finiteness in Using PRST     Index of Definitions   

This book is based on a three-term course of lectures that the author delivered at the Faculty of Computational Mathematics and Cybernetics of Moscow University. It is divided into three parts: Probability, Stochastic Processes and Statistics. The text includes approximately 200 problems and illustrative examples and is designed for university and college students, as well as anyone studying or using probability theory. Prof. G.P. Klimov, D.Sc. (Phys.-Math.), is a lecturer in mathematical statistics at Moscow University. His scientific interests lie within the field of random processes, queue theory, invariant statistical processes, and invariant statistical inference. He is the author of more than 60 papers and several books translated into German, Polish, and Bulgarian. Prof. G.P. Klimov has also delivered courses of lectures at the Sorbonne, Universities of Liège, Berlin, and the Stefan Banach International Mathematical Centre in Warsaw. Translated from the Russian by Oleg Efimov All credits to IA user  Quantum AI Okay, here’s the table of contents formatted for copy-pasteability into an RTF document. It includes bullet points to clearly delineate each section and subsection, making it easy to use within a document editor.     Table of Contents              Part 1: Probability Theory              Introduction          Chapter 1: Discrete Probability Space              Event and Probability              Probability Space              Independence of Events              Conditional Probability              Probability of Union of Events              Composite Probability Formula              Bayes Formula              Examples of Distributions              Random Variable, Mathematical Expectation              Independence of Random Variables              Additive and Multiplicative Properties of Mathematical Expectation              Moments              Variance Additivity for Independent RV’s              Generating Function              Binomial Distribution              Poisson Distribution              Negative Binomial Distribution              Approximation of Binomial by Poisson Distribution              Indicator of Event              Markov Inequality              Law of Large Numbers              Central Limit Theorem          Chapter 2: Probability Space              Definition of Probability Space              Construction of Probability Space              Complete Probability Space              Continuous Probability Space              Definition of Random Variable              Operations over RV’s              Equivalence and Almost Certain Convergence of RV's              Independence              Mathematical Expectation, Distribution Function, Distribution Density          Chapter 3: Subjective Probability (Supplement)              Semi-Markov Renewal Process          Part 2: Stochastic Processes              Introduction: Definition of Random Process, Kolmogorov Theorem on Construction of Processes from Partial Distributions          Chapter 3: Discrete- and Continuous-Time Markov Chains              Definition of Discrete- and Continuous-Time Markov Chains              Examples of Homogeneous Markov Chains              Limiting Distribution of Markov Chains          Chapter 4: Renewal Processes              Definition of Renewal Process              Elementary Renewal Theorem              Blackwell Theorem              Key Renewal Theorem          Chapter 5: Regenerative Processes              Definition of Regenerative Process              Limit Theorem for Regenerative Process              Law of Large Numbers for Regenerative Processes          Chapter 6: Elements of Random Analysis              Types of RV Convergence              Equivalent Definitions of RV Convergence and Basic Properties              Prokhorov Theorem and Its Applications              Stochastic Continuity              Stochastic Integrals of Non-Random Functions              Orthogonal Measure and Integral with Values in Hilbert Space              Differentiation and Integration of Mathematical Expectation of Processes          Chapter 7: Gaussian Processes              Definition of Gaussian Process              Wiener Process              Non-Homogeneous Wiener Process          Part 3: Statistics              Introduction          Chapter 9: Mathematical Statistics: Basic Concepts              Statistical Structure              Statistic              Sufficient Statistic              MVUE (Most Variable Unbiased Estimator)              Complete and Sufficient Statistic              Minimum Risk Criterion          Chapter 10: Point Estimation – Cramér-Rao Theory              MVUE (Most Variable Unbiased Estimator)              MVUE Uniqueness              Cramér-Rao Inequality              Efficient Estimators              Multidimensional Analogue of Cramér-Rao Inequality          Chapter 11: Point Estimation – Maximum Likelihood Method              Maximum Likelihood Estimator              Consistent Estimators              MLE Consistency              MLE Asymptotic Normality          Chapter 12: Point Estimation – Least Squares Method              Examples              Normal Equation              Gauss-Markov Theorem              Linear Regression Model              Normal Error Distribution          Chapter 13: Interval Estimation              Definition of Confidence Intervals              Likelihood Ratio Principle              Technique of Likelihood Ratio Principle Application          Chapter 14: Linear Model – Multidimensional Analysis              Introduction              Assumptions for Space of Parameters (6 = (n, a))              MLE Property              Estimating Function              Interval MLE for 6 = (n, a)              Independence of nm and a              Estimating pp and a Separately          Chapter 15: Statistical Hypothesis Testing              Hypothesis and Test              Power Function, Test Ordering              Fundamental Neyman-Pearson Lemma          Chapter 16: Sequential Analysis              Problem Posing              Probability Ratio Sequential Test for s Hypotheses              Average Sample Size Finiteness in Using PRST     Index of Definitions   

This is a survey of research in the theory of probability. The works of several scientists in this field are cited. The 'Markovian' process is discussed. The application of mathematics to the different scientific fields is cited.

This is a survey of research in the theory of probability. The works of several scientists in this field are cited. The 'Markovian' process is discussed. The application of mathematics to the different scientific fields is cited.