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Set, Measure and Probability Theory
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: River Publishers Series in Mathematical, Statistical and Computational Modelling for Engineering SeriesPublisher: Milton : River Publishers, 2023Description: 1 Online-Ressource (303 p.)ISBN:- 9781003808954
- 1003808956
- 9781032626451
- 1032626453
- 9781003808978
- 1003808972
- 511.322 23/eng/20240111
- QA248
Contents:
Summary: This book introduces the basic concepts of set theory, measure theory, the axiomatic theory of probability, random variables and multidimensional random variables, functions of random variables, convergence theorems, laws of large numbers, and fundamental inequalities. The idea is to present a seamless connection between the more abstract advanced set theory, the fundamental concepts from measure theory, and integration, to introduce the axiomatic theory of probability, filling in the gaps from previous books and leading to an interesting, robust and, hopefully, self-contained exposition of the theory. This book also presents an account of the historical evolution of probability theory as a mathematical discipline. Each chapter presents a short biography of the important scientists who helped develop the subject. Appendices include Fourier transforms in one and two dimensions, important formulas and inequalities and commented bibliography. Many examples, illustrations and graphics help the reader understand the theoryPPN: PPN: 1907962034Package identifier: Produktsigel: ZDB-4-NLEBK | BSZ-4-NLEBK-KAUB
Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Dedication -- Table of Contents -- Preface -- Acknowledgements -- List of Figures -- List of Tables -- List of Abbreviations -- Chapter 1: Advanced Set Theory -- 1.1: Set Theory -- 1.2: Basic Set Theory -- 1.3: The Axioms of Set Theory -- 1.4: Operations on Sets -- 1.5: Families of Sets -- 1.5.1: Indexing of Sets -- 1.6: An Algebra of Sets -- 1.7: The Borel Algebra -- 1.8: Cardinality -- 1.8.1: Equivalence of Sets -- 1.8.2: Countable Sets -- 1.8.3: Uncountable Sets -- 1.8.4: Cardinality Properties -- 1.9: Georg Cantor
1.10: Problems -- Chapter 2: Relations and Functions -- 2.1: Definition of a Relation -- 2.1.1: Relation Representation -- 2.1.2: Types of Relations -- 2.2: Definition of Function -- 2.2.1: Types of Functions -- 2.3: Mathematical Functions -- 2.3.1: Indicator Function -- 2.3.2: Fuzzy Sets -- 2.3.3: Properties of Set Functions -- 2.4: The Count of Arts and Mathematics -- 2.5: Problems -- Chapter 3: Fundamentals of Measure Theory -- 3.1: Measuring His -- 3.2: Measure in an Algebra of Sets -- 3.3: The Riemann Integral -- 3.4: The Lebesgue Integral -- 3.4.1: The Lebesgue Measure
3.4.2: Concept of the Lebesgue Integral -- 3.4.3: Properties of the Lebesgue Integral -- 3.5: Henri Lebesgue -- 3.6: Problems -- Chapter 4: Generalized Functions -- 4.1: A Note on Generalized Functions -- 4.2: The Unit Step Function -- 4.2.1: Properties of the Unit Step Function -- 4.3: The Signum Function -- 4.4: The Gate Function -- 4.5: The Impulse Function -- 4.5.1: The Functional -- 4.5.2: Properties of the Impulse Function -- 4.5.3: Composite Function with the Impulse -- 4.6: Doublet Generalized Function -- 4.7: The Ramp Function -- 4.8: The Exponential Function -- 4.9: Discrete Functions
4.9.1: Discrete Unit Step Function -- 4.9.2: Discrete Impulse Function -- 4.9.3: Discrete Ramp Function -- 4.10: Paul Dirac -- 4.11: Problems -- Chapter 5: Probability Theory -- 5.1: Reasoning in Games of Chance -- 5.2: Measurable Space -- 5.2.1: Probability Measure -- 5.2.2: Probability Measure with the Riemann Integral -- 5.2.3: Probability Measure with the Lebesgue Integral -- 5.3: The Axioms of Probability -- 5.4: Axioms of the Expectation Operator -- 5.5: Bayes' Theorem -- 5.6: Andrei Kolmogorov -- 5.7: Problems -- Chapter 6: Random Variables -- 6.1: The Concept of a Random Variable
6.1.1: Algebra Generated by a Random Variable -- 6.1.2: Lebesgue Measure and Probability -- 6.2: Cumulative Distribution Function -- 6.2.1: Change of Variable Theorem -- 6.3: Moments of a Random Variable -- 6.3.1: Properties Associated to the Expected Value -- 6.3.2: Definition of the Most Important Moments -- 6.4: Functions of Random Variables -- 6.4.1: General Formula for Transformation -- 6.5: Discrete Distributions -- 6.6: Characteristic Function -- 6.7: Conditional Distribution -- 6.8: Useful Distributions and Applications -- 6.9: Carl Friedrich Gauss -- 6.10: Problems
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