| Description |
1 online resource (240 pages) |
| Series |
Mathematical Engineering |
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Mathematical engineering
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| 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 |
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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 |
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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 |
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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 |
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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 |
| Summary |
The monograph compares two approaches that describe the statistical stability phenomenon - one proposed by the probability theory that ignores violations of statistical stability and another proposed by the theory of hyper-random phenomena that takes these violations into account. There are five parts. The first describes the phenomenon of statistical stability. The second outlines the mathematical foundations of probability theory. The third develops methods for detecting violations of statistical stability and presents the results of experimental research on actual processes of different physical nature that demonstrate the violations of statistical stability over broad observation intervals. The fourth part outlines the mathematical foundations of the theory of hyper-random phenomena. The fifth part discusses the problem of how to provide an adequate description of the world. The monograph should be interest to a wide readership: from university students on a first course majoring in physics, engineering, and mathematics to engineers, post-graduate students, and scientists carrying out research on the statistical laws of natural physical phenomena, developing and using statistical methods for high-precision measurement, prediction, and signal processing over broad observation intervals. To read the book, it is sufficient to be familiar with a standard first university course on mathematics |
| Bibliography |
Includes bibliographical references |
| Notes |
4.3 The Law of Large Numbers |
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Print version record |
| Subject |
Probabilities.
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Random variables.
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probability.
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Mensuration & systems of measurement.
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Probability & statistics.
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Statistical physics.
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Imaging systems & technology.
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Maths for engineers.
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MATHEMATICS -- Applied.
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MATHEMATICS -- Probability & Statistics -- General.
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Probabilities
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Random variables
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| Form |
Electronic book
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| ISBN |
9783319607801 |
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3319607804 |
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