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Table of Contents
Preface; References; Acknowledgments; References; Introduction; References; Contents; About the Author; Symbols; Part I: The Phenomenon of Statistical Stability; References; Chapter 1: The Physical Phenomenon of Statistical Stability; 1.1 Manifestation of the Phenomenon of Statistical Stability; 1.1.1 Statistical Stability of the Relative Frequency of Events; 1.1.2 Statistical Stability of Statistics; 1.2 Interpretations of the Phenomenon of Statistical Stability; 1.2.1 Perfect Statistical Stability; 1.2.2 Imperfect Statistical Stability.
1.3 Identical and Statistically Unpredictable Conditions1.4 Hilbertś Sixth Problem; 1.4.1 The Essence of the Problem; 1.4.2 Approaches to Axiomatizing Probability Theory; 1.4.3 How to Solve Hilbertś Sixth Problem; 1.5 Adequacy Axioms; 1.5.1 Description of the Phenomenon of Statistical Stability in the Framework of Probability Theory; 1.5.2 Description of the Phenomenon of Statistical Stability in the Framework of the Theory of Hyper-random Phenomena; 1.6 Is Probability a `Normal ́Physical Quantity?; References; Part II: Probability Theory; References; Chapter 2: Basis of Probability Theory.
2.1 The Concept of Random Phenomena2.2 Options for the Definition of Probability; 2.2.1 Classical Approach; 2.2.2 Statistical Approach; 2.2.3 Main Concepts of Set Theory; 2.2.4 Main Concepts of Measure Theory; 2.2.5 Axiomatic Definition of Probability; 2.2.6 Random Events; 2.3 Random Variables; 2.3.1 Basic Definitions; 2.3.2 Probabilistic Characteristics of a Scalar Random Variable; 2.3.3 Probabilistic Characteristics of a Discrete Random Variable; 2.3.4 Examples of Random Variables; 2.3.5 Numerical Parameters of Scalar Random Variables; 2.3.6 Numerical Parameters of Various Random Variables.
2.4 Vector Random Variables2.4.1 Probabilistic Characteristics of a System of Two Random Variables; 2.4.2 Numerical Parameters of a System of Two Random Variables; 2.4.3 System of Two Jointly Gaussian Random Variables; 2.4.4 Characteristics and Parameters of a System of more than Two Random Variables; 2.5 Operations on Random Variables; References; Chapter 3: Stochastic Functions; 3.1 Main Concepts; 3.2 Description of Stochastic Processes; 3.3 Gaussian Stochastic Process; 3.4 Stationary Stochastic Processes; 3.4.1 Stochastic Processes That Are Stationary in the Narrow Sense.
3.4.2 Stochastic Processes That Are Stationary in the Broad Sense3.5 Spectral Description of Stochastic Processes; 3.5.1 Wiener-Khinchin Transformation; 3.5.2 Narrowband and Broadband Processes; 3.5.3 Generalized Wiener-Khinchin Transformation; 3.6 Ergodic Stochastic Processes; 3.7 Transformation of Stochastic Processes; References; Chapter 4: Fundamentals of the Mathematical Statistics of Probability Theory; 4.1 Statistics of Random Variables; 4.1.1 A Random Sample; 4.1.2 Assessments of Probability Characteristics; 4.1.3 Assessment of Moments; 4.2 Convergence of Sequences of Random Variables; 4.3 The Law of Large Numbers.
1.3 Identical and Statistically Unpredictable Conditions1.4 Hilbertś Sixth Problem; 1.4.1 The Essence of the Problem; 1.4.2 Approaches to Axiomatizing Probability Theory; 1.4.3 How to Solve Hilbertś Sixth Problem; 1.5 Adequacy Axioms; 1.5.1 Description of the Phenomenon of Statistical Stability in the Framework of Probability Theory; 1.5.2 Description of the Phenomenon of Statistical Stability in the Framework of the Theory of Hyper-random Phenomena; 1.6 Is Probability a `Normal ́Physical Quantity?; References; Part II: Probability Theory; References; Chapter 2: Basis of Probability Theory.
2.1 The Concept of Random Phenomena2.2 Options for the Definition of Probability; 2.2.1 Classical Approach; 2.2.2 Statistical Approach; 2.2.3 Main Concepts of Set Theory; 2.2.4 Main Concepts of Measure Theory; 2.2.5 Axiomatic Definition of Probability; 2.2.6 Random Events; 2.3 Random Variables; 2.3.1 Basic Definitions; 2.3.2 Probabilistic Characteristics of a Scalar Random Variable; 2.3.3 Probabilistic Characteristics of a Discrete Random Variable; 2.3.4 Examples of Random Variables; 2.3.5 Numerical Parameters of Scalar Random Variables; 2.3.6 Numerical Parameters of Various Random Variables.
2.4 Vector Random Variables2.4.1 Probabilistic Characteristics of a System of Two Random Variables; 2.4.2 Numerical Parameters of a System of Two Random Variables; 2.4.3 System of Two Jointly Gaussian Random Variables; 2.4.4 Characteristics and Parameters of a System of more than Two Random Variables; 2.5 Operations on Random Variables; References; Chapter 3: Stochastic Functions; 3.1 Main Concepts; 3.2 Description of Stochastic Processes; 3.3 Gaussian Stochastic Process; 3.4 Stationary Stochastic Processes; 3.4.1 Stochastic Processes That Are Stationary in the Narrow Sense.
3.4.2 Stochastic Processes That Are Stationary in the Broad Sense3.5 Spectral Description of Stochastic Processes; 3.5.1 Wiener-Khinchin Transformation; 3.5.2 Narrowband and Broadband Processes; 3.5.3 Generalized Wiener-Khinchin Transformation; 3.6 Ergodic Stochastic Processes; 3.7 Transformation of Stochastic Processes; References; Chapter 4: Fundamentals of the Mathematical Statistics of Probability Theory; 4.1 Statistics of Random Variables; 4.1.1 A Random Sample; 4.1.2 Assessments of Probability Characteristics; 4.1.3 Assessment of Moments; 4.2 Convergence of Sequences of Random Variables; 4.3 The Law of Large Numbers.