Set, measure and probability theory:
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 seaml...
Gespeichert in:
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Gistrup, Denmark
River Publishers
[2023]
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| Schriftenreihe: | River Publishers series in mathematical, statistical and computational modelling for engineering
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| Schlagworte: | |
| Online-Zugang: | DE-573 Volltext |
| Zusammenfassung: | 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 theory |
| Beschreibung: | 1 Online-Ressource (xxv, 376 Seiten) Illustrationen, Diagramme |
| ISBN: | 9788770228824 8770228825 |
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| 100 | 1 | |a Alencar, Marcelos S. |d 1957- |e Verfasser |0 (DE-588)1146738382 |4 aut | |
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| 264 | 1 | |a Gistrup, Denmark |b River Publishers |c [2023] | |
| 300 | |a 1 Online-Ressource (xxv, 376 Seiten) |b Illustrationen, Diagramme | ||
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| 505 | 8 | |a Preface xiii Acknowledgements xv List of Figures xvii List of Tables xxiii List of Abbreviations xxv 1 Advanced Set Theory 1 1.1 Set Theory 1 1.2 Basic Set Theory 2 1.3 The Axioms of Set Theory 4 1.4 Operations on Sets 5 1.5 Families of Sets 7 1.5.1 Indexing of Sets 10 1.6 An Algebra of Sets 10 1.7 The Borel Algebra 12 1.8 Cardinality 13 1.8.1 Equivalence of Sets 13 1.8.2 Countable Sets 14 1.8.3 Uncountable Sets 16 1.8.4 Cardinality Properties 19 1.9 Georg Cantor 20 1.10 Problems 22 2 Relations and Functions 23 2.1 Definition of a Relation 23 2.1.1 Relation Representation 24 2.1.2 Types of Relations 26 2.2 Definition of Function 27 2.2.1 Types of Functions 28 2.3 Mathematical Functions 31 2.3.1 Indicator Function 33 2.3.2 Fuzzy Sets 34 2.3.3 Properties of Set Functions 36 2.4 The Count of Arts and Mathematics 38 2.5 Problems 40 3 Fundamentals of Measure Theory 41 3.1 Measuring History 41 3.2 Measure in an Algebra of Sets 44 3.3 The Riemann Integral 47 3.4 The Lebesgue Integral 54 | |
| 505 | 8 | |a 3.4.1 The Lebesgue Measure 57 3.4.2 Concept of the Lebesgue Integral 61 3.4.3 Properties of the Lebesgue Integral 62 3.5 Henri Lebesgue 66 3.6 Problems 67 4 Generalized Functions 69 4.1 A Note on Generalized Functions 69 4.2 The Unit Step Function 70 4.2.1 Properties of the Unit Step Function 71 4.3 The Signum Function 72 4.4 The Gate Function 73 4.5 The Impulse Function 73 4.5.1 The Functional 79 4.5.2 Properties of the Impulse Function 81 4.5.3 Composite Function with the Impulse 82 4.6 Doublet Generalized Function 85 4.7 The Ramp Function 86 4.8 The Exponential Function 87 4.9 Discrete Functions 89 4.9.1 Discrete Unit Step Function 90 4.9.2 Discrete Impulse Function 90 4.9.3 Discrete Ramp Function 91 4.10 Paul Dirac 94 4.11 Problems 95 5 Probability Theory 97 5.1 Reasoning in Games of Chance 97 5.2 Measurable Space 99 5.2.1 Probability Measure 100 5.2.2 Probability Measure with the Riemann Integral 100 5.2.3 Probability Measure with the Lebesgue Integral 101 5.3 The Axioms of | |
| 505 | 8 | |a Probability 105 5.4 Axioms of the Expectation Operator 108 5.5 Bayes’ Theorem 109 5.6 Andrei Kolmogorov 113 5.7 Problems 114 6 Random Variables 117 6.1 The Concept of a Random Variable 117 6.1.1 Algebra Generated by a Random Variable 119 6.1.2 Lebesgue Measure and Probability 119 6.2 Cumulative Distribution Function 120 6.2.1 Change of Variable Theorem 126 6.3 Moments of a Random Variable 127 6.3.1 Properties Associated to the Expected Value 128 6.3.2 Definition of the Most Important Moments 129 6.4 Functions of Random Variables 132 6.4.1 General Formula for Transformation 138 6.5 Discrete Distributions 144 6.6 Characteristic Function 149 6.7 Conditional Distribution 151 6.8 Useful Distributions and Applications 156 6.9 Carl Friedrich Gauss 164 6.10 Problems 165 7 Joint Random Variables 167 7.1 An Extension of the Concept of Random Variable 167 7.2 Properties of Probability Distributions 171 7.3 Moments in Two Dimensions 172 7.4 Conditional Moments 177 7.5 Two-Dimensional | |
| 505 | 8 | |a Characteristic Function 178 7.5.1 Sum of Random Variables 180 7.6 Function of Joint Random Variables 181 7.7 Transformation of Random Vectors 186 7.8 Complex Random Variables 188 7.9 Félix Borel 190 7.10 Problems 191 8 Probability Fundamental Inequalities 193 8.1 Historical Notes 193 8.2 Tchebychev’s Inequality 195 8.3 Markov’s Inequality 197 8.4 Bienaymé’s Inequality 198 8.5 Jensen’s Inequality 198 8.6 Chernoff’s Inequality 199 8.7 Kolmogorov’s Inequality 201 8.8 Schwarz’ Inequality 202 8.9 Hḻder’s Inequality 203 8.10 Lyapunov’s Inequality 204 8.11 Minkowsky’s Inequality 204 8.12 Fatou’s Lemma 205 8.13 About Arguments and Proofs 207 8.14 Problems 208 9 Convergence and the Law of Large Numbers 211 9.1 Forms of Convergence in Probability Theory 211 9.2 Types of Convergence 211 9.2.1 Convergence in Probability 212 9.2.2 Almost Sure Convergence 212 9.2.3 Sure Convergence 213 9.2.4 Convergence in Distribution | |
| 505 | 8 | |a 214 9.2.5 Convergence in Mean of Order r 214 9.2.6 Convergence in Mean 214 9.2.7 Convergence in Mean Square 215 9.2.8 Convergence in Measure 216 9.3 Relationships Between the Types of Convergence 216 9.4 Weak Law of Large Numbers 217 9.5 Strong Law of Large Numbers 219 9.6 Central Limit Theorem 222 9.6.1 Demonstration of the Theorem 222 9.6.2 Central Limit Theorem for Products 225 9.7 Pierre-Simon Laplace 226 9.8 Problems 227 A Formulas and Important Inequalities 231 B Fourier Transform 239 B.1 Table of Fourier Transforms 239 C Commented Bibliography 247 Bibliography 259 Index 265 About the Authors 275 | |
| 520 | 3 | |a 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 theory | |
| 653 | 0 | |a Set theory | |
| 653 | 0 | |a Measure theory | |
| 653 | 0 | |a Probabilities | |
| 653 | 0 | |a Théorie des ensembles | |
| 653 | 0 | |a Théorie de la mesure | |
| 653 | 0 | |a Probabilités | |
| 653 | 0 | |a probability | |
| 653 | 0 | |a Measure theory | |
| 653 | 0 | |a Set theory | |
| 700 | 1 | |a Alencar, Raphael Tavares de |e Sonstige |4 oth | |
| 776 | 0 | 8 | |i Print version |a ALENCAR, MARCELO S.. ALENCAR, RAPHAEL T. |t SET, MEASURE AND PROBABILITY THEORY. |d Gistrup, Denmark: RIVER PUBLISHERS, 2023 |z 8770228477 |
| 856 | 4 | 0 | |u https://ieeexplore.ieee.org/book/10301704 |x Aggregator |z URL des Erstveröffentlichers |3 Volltext |
| 912 | |a ZDB-37-RPEB | ||
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-034848014 | |
| 966 | e | |u https://ieeexplore.ieee.org/book/10301704 |l DE-573 |p ZDB-37-RPEB |x Verlag |3 Volltext | |
Datensatz im Suchindex
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| contents | Preface xiii Acknowledgements xv List of Figures xvii List of Tables xxiii List of Abbreviations xxv 1 Advanced Set Theory 1 1.1 Set Theory 1 1.2 Basic Set Theory 2 1.3 The Axioms of Set Theory 4 1.4 Operations on Sets 5 1.5 Families of Sets 7 1.5.1 Indexing of Sets 10 1.6 An Algebra of Sets 10 1.7 The Borel Algebra 12 1.8 Cardinality 13 1.8.1 Equivalence of Sets 13 1.8.2 Countable Sets 14 1.8.3 Uncountable Sets 16 1.8.4 Cardinality Properties 19 1.9 Georg Cantor 20 1.10 Problems 22 2 Relations and Functions 23 2.1 Definition of a Relation 23 2.1.1 Relation Representation 24 2.1.2 Types of Relations 26 2.2 Definition of Function 27 2.2.1 Types of Functions 28 2.3 Mathematical Functions 31 2.3.1 Indicator Function 33 2.3.2 Fuzzy Sets 34 2.3.3 Properties of Set Functions 36 2.4 The Count of Arts and Mathematics 38 2.5 Problems 40 3 Fundamentals of Measure Theory 41 3.1 Measuring History 41 3.2 Measure in an Algebra of Sets 44 3.3 The Riemann Integral 47 3.4 The Lebesgue Integral 54 3.4.1 The Lebesgue Measure 57 3.4.2 Concept of the Lebesgue Integral 61 3.4.3 Properties of the Lebesgue Integral 62 3.5 Henri Lebesgue 66 3.6 Problems 67 4 Generalized Functions 69 4.1 A Note on Generalized Functions 69 4.2 The Unit Step Function 70 4.2.1 Properties of the Unit Step Function 71 4.3 The Signum Function 72 4.4 The Gate Function 73 4.5 The Impulse Function 73 4.5.1 The Functional 79 4.5.2 Properties of the Impulse Function 81 4.5.3 Composite Function with the Impulse 82 4.6 Doublet Generalized Function 85 4.7 The Ramp Function 86 4.8 The Exponential Function 87 4.9 Discrete Functions 89 4.9.1 Discrete Unit Step Function 90 4.9.2 Discrete Impulse Function 90 4.9.3 Discrete Ramp Function 91 4.10 Paul Dirac 94 4.11 Problems 95 5 Probability Theory 97 5.1 Reasoning in Games of Chance 97 5.2 Measurable Space 99 5.2.1 Probability Measure 100 5.2.2 Probability Measure with the Riemann Integral 100 5.2.3 Probability Measure with the Lebesgue Integral 101 5.3 The Axioms of Probability 105 5.4 Axioms of the Expectation Operator 108 5.5 Bayes’ Theorem 109 5.6 Andrei Kolmogorov 113 5.7 Problems 114 6 Random Variables 117 6.1 The Concept of a Random Variable 117 6.1.1 Algebra Generated by a Random Variable 119 6.1.2 Lebesgue Measure and Probability 119 6.2 Cumulative Distribution Function 120 6.2.1 Change of Variable Theorem 126 6.3 Moments of a Random Variable 127 6.3.1 Properties Associated to the Expected Value 128 6.3.2 Definition of the Most Important Moments 129 6.4 Functions of Random Variables 132 6.4.1 General Formula for Transformation 138 6.5 Discrete Distributions 144 6.6 Characteristic Function 149 6.7 Conditional Distribution 151 6.8 Useful Distributions and Applications 156 6.9 Carl Friedrich Gauss 164 6.10 Problems 165 7 Joint Random Variables 167 7.1 An Extension of the Concept of Random Variable 167 7.2 Properties of Probability Distributions 171 7.3 Moments in Two Dimensions 172 7.4 Conditional Moments 177 7.5 Two-Dimensional Characteristic Function 178 7.5.1 Sum of Random Variables 180 7.6 Function of Joint Random Variables 181 7.7 Transformation of Random Vectors 186 7.8 Complex Random Variables 188 7.9 Félix Borel 190 7.10 Problems 191 8 Probability Fundamental Inequalities 193 8.1 Historical Notes 193 8.2 Tchebychev’s Inequality 195 8.3 Markov’s Inequality 197 8.4 Bienaymé’s Inequality 198 8.5 Jensen’s Inequality 198 8.6 Chernoff’s Inequality 199 8.7 Kolmogorov’s Inequality 201 8.8 Schwarz’ Inequality 202 8.9 Hḻder’s Inequality 203 8.10 Lyapunov’s Inequality 204 8.11 Minkowsky’s Inequality 204 8.12 Fatou’s Lemma 205 8.13 About Arguments and Proofs 207 8.14 Problems 208 9 Convergence and the Law of Large Numbers 211 9.1 Forms of Convergence in Probability Theory 211 9.2 Types of Convergence 211 9.2.1 Convergence in Probability 212 9.2.2 Almost Sure Convergence 212 9.2.3 Sure Convergence 213 9.2.4 Convergence in Distribution 214 9.2.5 Convergence in Mean of Order r 214 9.2.6 Convergence in Mean 214 9.2.7 Convergence in Mean Square 215 9.2.8 Convergence in Measure 216 9.3 Relationships Between the Types of Convergence 216 9.4 Weak Law of Large Numbers 217 9.5 Strong Law of Large Numbers 219 9.6 Central Limit Theorem 222 9.6.1 Demonstration of the Theorem 222 9.6.2 Central Limit Theorem for Products 225 9.7 Pierre-Simon Laplace 226 9.8 Problems 227 A Formulas and Important Inequalities 231 B Fourier Transform 239 B.1 Table of Fourier Transforms 239 C Commented Bibliography 247 Bibliography 259 Index 265 About the Authors 275 |
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| id | DE-604.BV049502917 |
| illustrated | Not Illustrated |
| index_date | 2024-07-03T23:21:44Z |
| indexdate | 2024-07-31T00:20:22Z |
| institution | BVB |
| isbn | 9788770228824 8770228825 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-034848014 |
| oclc_num | 1418706627 |
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| owner | DE-573 |
| owner_facet | DE-573 |
| physical | 1 Online-Ressource (xxv, 376 Seiten) Illustrationen, Diagramme |
| psigel | ZDB-37-RPEB |
| publishDate | 2023 |
| publishDateSearch | 2023 |
| publishDateSort | 2023 |
| publisher | River Publishers |
| record_format | marc |
| series2 | River Publishers series in mathematical, statistical and computational modelling for engineering |
| spelling | Alencar, Marcelos S. 1957- Verfasser (DE-588)1146738382 aut Set, measure and probability theory Marcelo S. Alencar, Raphael T. Alencar Gistrup, Denmark River Publishers [2023] 1 Online-Ressource (xxv, 376 Seiten) Illustrationen, Diagramme txt rdacontent c rdamedia cr rdacarrier River Publishers series in mathematical, statistical and computational modelling for engineering Preface xiii Acknowledgements xv List of Figures xvii List of Tables xxiii List of Abbreviations xxv 1 Advanced Set Theory 1 1.1 Set Theory 1 1.2 Basic Set Theory 2 1.3 The Axioms of Set Theory 4 1.4 Operations on Sets 5 1.5 Families of Sets 7 1.5.1 Indexing of Sets 10 1.6 An Algebra of Sets 10 1.7 The Borel Algebra 12 1.8 Cardinality 13 1.8.1 Equivalence of Sets 13 1.8.2 Countable Sets 14 1.8.3 Uncountable Sets 16 1.8.4 Cardinality Properties 19 1.9 Georg Cantor 20 1.10 Problems 22 2 Relations and Functions 23 2.1 Definition of a Relation 23 2.1.1 Relation Representation 24 2.1.2 Types of Relations 26 2.2 Definition of Function 27 2.2.1 Types of Functions 28 2.3 Mathematical Functions 31 2.3.1 Indicator Function 33 2.3.2 Fuzzy Sets 34 2.3.3 Properties of Set Functions 36 2.4 The Count of Arts and Mathematics 38 2.5 Problems 40 3 Fundamentals of Measure Theory 41 3.1 Measuring History 41 3.2 Measure in an Algebra of Sets 44 3.3 The Riemann Integral 47 3.4 The Lebesgue Integral 54 3.4.1 The Lebesgue Measure 57 3.4.2 Concept of the Lebesgue Integral 61 3.4.3 Properties of the Lebesgue Integral 62 3.5 Henri Lebesgue 66 3.6 Problems 67 4 Generalized Functions 69 4.1 A Note on Generalized Functions 69 4.2 The Unit Step Function 70 4.2.1 Properties of the Unit Step Function 71 4.3 The Signum Function 72 4.4 The Gate Function 73 4.5 The Impulse Function 73 4.5.1 The Functional 79 4.5.2 Properties of the Impulse Function 81 4.5.3 Composite Function with the Impulse 82 4.6 Doublet Generalized Function 85 4.7 The Ramp Function 86 4.8 The Exponential Function 87 4.9 Discrete Functions 89 4.9.1 Discrete Unit Step Function 90 4.9.2 Discrete Impulse Function 90 4.9.3 Discrete Ramp Function 91 4.10 Paul Dirac 94 4.11 Problems 95 5 Probability Theory 97 5.1 Reasoning in Games of Chance 97 5.2 Measurable Space 99 5.2.1 Probability Measure 100 5.2.2 Probability Measure with the Riemann Integral 100 5.2.3 Probability Measure with the Lebesgue Integral 101 5.3 The Axioms of Probability 105 5.4 Axioms of the Expectation Operator 108 5.5 Bayes’ Theorem 109 5.6 Andrei Kolmogorov 113 5.7 Problems 114 6 Random Variables 117 6.1 The Concept of a Random Variable 117 6.1.1 Algebra Generated by a Random Variable 119 6.1.2 Lebesgue Measure and Probability 119 6.2 Cumulative Distribution Function 120 6.2.1 Change of Variable Theorem 126 6.3 Moments of a Random Variable 127 6.3.1 Properties Associated to the Expected Value 128 6.3.2 Definition of the Most Important Moments 129 6.4 Functions of Random Variables 132 6.4.1 General Formula for Transformation 138 6.5 Discrete Distributions 144 6.6 Characteristic Function 149 6.7 Conditional Distribution 151 6.8 Useful Distributions and Applications 156 6.9 Carl Friedrich Gauss 164 6.10 Problems 165 7 Joint Random Variables 167 7.1 An Extension of the Concept of Random Variable 167 7.2 Properties of Probability Distributions 171 7.3 Moments in Two Dimensions 172 7.4 Conditional Moments 177 7.5 Two-Dimensional Characteristic Function 178 7.5.1 Sum of Random Variables 180 7.6 Function of Joint Random Variables 181 7.7 Transformation of Random Vectors 186 7.8 Complex Random Variables 188 7.9 Félix Borel 190 7.10 Problems 191 8 Probability Fundamental Inequalities 193 8.1 Historical Notes 193 8.2 Tchebychev’s Inequality 195 8.3 Markov’s Inequality 197 8.4 Bienaymé’s Inequality 198 8.5 Jensen’s Inequality 198 8.6 Chernoff’s Inequality 199 8.7 Kolmogorov’s Inequality 201 8.8 Schwarz’ Inequality 202 8.9 Hḻder’s Inequality 203 8.10 Lyapunov’s Inequality 204 8.11 Minkowsky’s Inequality 204 8.12 Fatou’s Lemma 205 8.13 About Arguments and Proofs 207 8.14 Problems 208 9 Convergence and the Law of Large Numbers 211 9.1 Forms of Convergence in Probability Theory 211 9.2 Types of Convergence 211 9.2.1 Convergence in Probability 212 9.2.2 Almost Sure Convergence 212 9.2.3 Sure Convergence 213 9.2.4 Convergence in Distribution 214 9.2.5 Convergence in Mean of Order r 214 9.2.6 Convergence in Mean 214 9.2.7 Convergence in Mean Square 215 9.2.8 Convergence in Measure 216 9.3 Relationships Between the Types of Convergence 216 9.4 Weak Law of Large Numbers 217 9.5 Strong Law of Large Numbers 219 9.6 Central Limit Theorem 222 9.6.1 Demonstration of the Theorem 222 9.6.2 Central Limit Theorem for Products 225 9.7 Pierre-Simon Laplace 226 9.8 Problems 227 A Formulas and Important Inequalities 231 B Fourier Transform 239 B.1 Table of Fourier Transforms 239 C Commented Bibliography 247 Bibliography 259 Index 265 About the Authors 275 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 theory Set theory Measure theory Probabilities Théorie des ensembles Théorie de la mesure Probabilités probability Alencar, Raphael Tavares de Sonstige oth Print version ALENCAR, MARCELO S.. ALENCAR, RAPHAEL T. SET, MEASURE AND PROBABILITY THEORY. Gistrup, Denmark: RIVER PUBLISHERS, 2023 8770228477 https://ieeexplore.ieee.org/book/10301704 Aggregator URL des Erstveröffentlichers Volltext |
| spellingShingle | Alencar, Marcelos S. 1957- Set, measure and probability theory Preface xiii Acknowledgements xv List of Figures xvii List of Tables xxiii List of Abbreviations xxv 1 Advanced Set Theory 1 1.1 Set Theory 1 1.2 Basic Set Theory 2 1.3 The Axioms of Set Theory 4 1.4 Operations on Sets 5 1.5 Families of Sets 7 1.5.1 Indexing of Sets 10 1.6 An Algebra of Sets 10 1.7 The Borel Algebra 12 1.8 Cardinality 13 1.8.1 Equivalence of Sets 13 1.8.2 Countable Sets 14 1.8.3 Uncountable Sets 16 1.8.4 Cardinality Properties 19 1.9 Georg Cantor 20 1.10 Problems 22 2 Relations and Functions 23 2.1 Definition of a Relation 23 2.1.1 Relation Representation 24 2.1.2 Types of Relations 26 2.2 Definition of Function 27 2.2.1 Types of Functions 28 2.3 Mathematical Functions 31 2.3.1 Indicator Function 33 2.3.2 Fuzzy Sets 34 2.3.3 Properties of Set Functions 36 2.4 The Count of Arts and Mathematics 38 2.5 Problems 40 3 Fundamentals of Measure Theory 41 3.1 Measuring History 41 3.2 Measure in an Algebra of Sets 44 3.3 The Riemann Integral 47 3.4 The Lebesgue Integral 54 3.4.1 The Lebesgue Measure 57 3.4.2 Concept of the Lebesgue Integral 61 3.4.3 Properties of the Lebesgue Integral 62 3.5 Henri Lebesgue 66 3.6 Problems 67 4 Generalized Functions 69 4.1 A Note on Generalized Functions 69 4.2 The Unit Step Function 70 4.2.1 Properties of the Unit Step Function 71 4.3 The Signum Function 72 4.4 The Gate Function 73 4.5 The Impulse Function 73 4.5.1 The Functional 79 4.5.2 Properties of the Impulse Function 81 4.5.3 Composite Function with the Impulse 82 4.6 Doublet Generalized Function 85 4.7 The Ramp Function 86 4.8 The Exponential Function 87 4.9 Discrete Functions 89 4.9.1 Discrete Unit Step Function 90 4.9.2 Discrete Impulse Function 90 4.9.3 Discrete Ramp Function 91 4.10 Paul Dirac 94 4.11 Problems 95 5 Probability Theory 97 5.1 Reasoning in Games of Chance 97 5.2 Measurable Space 99 5.2.1 Probability Measure 100 5.2.2 Probability Measure with the Riemann Integral 100 5.2.3 Probability Measure with the Lebesgue Integral 101 5.3 The Axioms of Probability 105 5.4 Axioms of the Expectation Operator 108 5.5 Bayes’ Theorem 109 5.6 Andrei Kolmogorov 113 5.7 Problems 114 6 Random Variables 117 6.1 The Concept of a Random Variable 117 6.1.1 Algebra Generated by a Random Variable 119 6.1.2 Lebesgue Measure and Probability 119 6.2 Cumulative Distribution Function 120 6.2.1 Change of Variable Theorem 126 6.3 Moments of a Random Variable 127 6.3.1 Properties Associated to the Expected Value 128 6.3.2 Definition of the Most Important Moments 129 6.4 Functions of Random Variables 132 6.4.1 General Formula for Transformation 138 6.5 Discrete Distributions 144 6.6 Characteristic Function 149 6.7 Conditional Distribution 151 6.8 Useful Distributions and Applications 156 6.9 Carl Friedrich Gauss 164 6.10 Problems 165 7 Joint Random Variables 167 7.1 An Extension of the Concept of Random Variable 167 7.2 Properties of Probability Distributions 171 7.3 Moments in Two Dimensions 172 7.4 Conditional Moments 177 7.5 Two-Dimensional Characteristic Function 178 7.5.1 Sum of Random Variables 180 7.6 Function of Joint Random Variables 181 7.7 Transformation of Random Vectors 186 7.8 Complex Random Variables 188 7.9 Félix Borel 190 7.10 Problems 191 8 Probability Fundamental Inequalities 193 8.1 Historical Notes 193 8.2 Tchebychev’s Inequality 195 8.3 Markov’s Inequality 197 8.4 Bienaymé’s Inequality 198 8.5 Jensen’s Inequality 198 8.6 Chernoff’s Inequality 199 8.7 Kolmogorov’s Inequality 201 8.8 Schwarz’ Inequality 202 8.9 Hḻder’s Inequality 203 8.10 Lyapunov’s Inequality 204 8.11 Minkowsky’s Inequality 204 8.12 Fatou’s Lemma 205 8.13 About Arguments and Proofs 207 8.14 Problems 208 9 Convergence and the Law of Large Numbers 211 9.1 Forms of Convergence in Probability Theory 211 9.2 Types of Convergence 211 9.2.1 Convergence in Probability 212 9.2.2 Almost Sure Convergence 212 9.2.3 Sure Convergence 213 9.2.4 Convergence in Distribution 214 9.2.5 Convergence in Mean of Order r 214 9.2.6 Convergence in Mean 214 9.2.7 Convergence in Mean Square 215 9.2.8 Convergence in Measure 216 9.3 Relationships Between the Types of Convergence 216 9.4 Weak Law of Large Numbers 217 9.5 Strong Law of Large Numbers 219 9.6 Central Limit Theorem 222 9.6.1 Demonstration of the Theorem 222 9.6.2 Central Limit Theorem for Products 225 9.7 Pierre-Simon Laplace 226 9.8 Problems 227 A Formulas and Important Inequalities 231 B Fourier Transform 239 B.1 Table of Fourier Transforms 239 C Commented Bibliography 247 Bibliography 259 Index 265 About the Authors 275 |
| title | Set, measure and probability theory |
| title_auth | Set, measure and probability theory |
| title_exact_search | Set, measure and probability theory |
| title_exact_search_txtP | Set, measure and probability theory |
| title_full | Set, measure and probability theory Marcelo S. Alencar, Raphael T. Alencar |
| title_fullStr | Set, measure and probability theory Marcelo S. Alencar, Raphael T. Alencar |
| title_full_unstemmed | Set, measure and probability theory Marcelo S. Alencar, Raphael T. Alencar |
| title_short | Set, measure and probability theory |
| title_sort | set measure and probability theory |
| url | https://ieeexplore.ieee.org/book/10301704 |
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