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Informationen zum Autor IONUT FLORESCU, PhD, is Research Associate Professor of Financial Engineering and Director of the Hanlon Financial Systems Lab at Stevens Institute of Technology. He has published extensively in his areas of research interest, which include stochastic volatility, stochastic partial differential equations, Monte Carlo methods, and numerical methods for stochastic processes. CIPRIAN A. TUDOR, PhD, is Professor of Mathematics at the University of Lille 1, France. His research interests include Brownian motion, limit theorems, statistical inference for stochastic processes, and financial mathematics. He has over eighty scientific publications in various internationally recognized journals on probability theory and statistics. He serves as a referee for over a dozen journals and has spoken at more than thirty-five conferences worldwide. Klappentext THE COMPLETE COLLECTION NECESSARY FOR A CONCRETE UNDERSTANDING OF PROBABILITY Written in a clear, accessible, and comprehensive manner, the Handbook of Probability presents the fundamentals of probability with an emphasis on the balance of theory, application, and methodology. Utilizing basic examples throughout, the handbook expertly transitions between concepts and practice to allow readers an inclusive introduction to the field of probability. The book provides a useful format with self-contained chapters, allowing the reader easy and quick reference. Each chapter includes an introduction, historical background, theory and applications, algorithms, and exercises. The Handbook of Probability offers coverage of: Probability Space Probability Measure Random Variables Random Vectors in Rn Characteristic Function Moment Generating Function Gaussian Random Vectors Convergence Types Limit Theorems The Handbook of Probability is an ideal resource for researchers and practitioners in numerous fields, such as mathematics, statistics, operations research, engineering, medicine, and finance, as well as a useful text for graduate students. Zusammenfassung This handbook provides a complete, but accessible compendium of all the major theorems, applications, and methodologies that are necessary for a clear understanding of probability. Each chapter is self-contained utilizing a common format. Algorithms and formulae are stressed when necessary and in an easy-to-locate fashion. Inhaltsverzeichnis List of Figures xv Preface xvii Introduction xix 1 Probability Space 1 1.1 Introduction/Purpose of the Chapter 1 1.2 Vignette/Historical Notes 2 1.3 Notations and Definitions 2 1.4 Theory and Applications 4 1.4.1 Algebras 4 1.4.2 Sigma Algebras 5 1.4.3 Measurable Spaces 7 1.4.4 Examples 7 1.4.5 The Borel _-Algebra 9 1.5 Summary 12 Exercises 12 2 Probability Measure 15 2.1 Introduction/Purpose of the Chapter 15 2.2 Vignette/Historical Notes 16 2.3 Theory and Applications 17 2.3.1 Definition and Basic Properties 17 2.3.2 Uniqueness of Probability Measures 22 2.3.3 Monotone Class 24 2.3.4 Examples 26 2.3.5 Monotone Convergence Properties of Probability 28 2.3.6 Conditional Probability 31 2.3.7 Independence of Events and _-Fields 39 2.3.8 Borel-Cantelli Lemmas 46 2.3.9 Fatou's Lemmas 48 2.3.10 Kolmogorov's Zero-One Law 49 2.4 Lebesgue Measure on the Unit Interval (01] 50 Exercises 52 3 Random Variables: Generalities 63 3.1 Introduction/Purpose of the Chapter 63 3.2 Vignette/Historical Notes 63 3.3 Theory and Applications 64 3.3.1 Definition 64 3.3....
List of contents
List of Figures xv
Preface xvii
Introduction xix
1 Probability Space 1
1.1 Introduction/Purpose of the Chapter 1
1.2 Vignette/Historical Notes 2
1.3 Notations and Definitions 2
1.4 Theory and Applications 4
1.4.1 Algebras 4
1.4.2 Sigma Algebras 5
1.4.3 Measurable Spaces 7
1.4.4 Examples 7
1.4.5 The Borel sigma-Algebra 9
1.5 Summary 12
Exercises 12
2 Probability Measure 15
2.1 Introduction/Purpose of the Chapter 15
2.2 Vignette/Historical Notes 16
2.3 Theory and Applications 17
2.3.1 Definition and Basic Properties 17
2.3.2 Uniqueness of Probability Measures 22
2.3.3 Monotone Class 24
2.3.4 Examples 26
2.3.5 Monotone Convergence Properties of Probability 28
2.3.6 Conditional Probability 31
2.3.7 Independence of Events and sigma-Fields 39
2.3.8 Borel-Cantelli Lemmas 46
2.3.9 Fatou's Lemmas 48
2.3.10 Kolmogorov's Zero-One Law 49
2.4 Lebesgue Measure on the Unit Interval (01] 50
Exercises 52
3 Random Variables: Generalities 63
3.1 Introduction/Purpose of the Chapter 63
3.2 Vignette/Historical Notes 63
3.3 Theory and Applications 64
3.3.1 Definition 64
3.3.2 The Distribution of a Random Variable 65
3.3.3 The Cumulative Distribution Function of a Random Variable 67
3.3.4 Independence of Random Variables 70
Exercises 71
4 Random Variables: The Discrete Case 79
4.1 Introduction/Purpose of the Chapter 79
4.2 Vignette/Historical Notes 80
4.3 Theory and Applications 80
4.3.1 Definition and Basic Facts 80
4.3.2 Moments 84
4.4 Examples of Discrete Random Variables 89
4.4.1 The (Discrete) Uniform Distribution 89
4.4.2 Bernoulli Distribution 91
4.4.3 Binomial (n p) Distribution 92
4.4.4 Geometric (p) Distribution 95
4.4.5 Negative Binomial (r p) Distribution 101
4.4.6 Hypergeometric Distribution (N m n) 102
4.4.7 Poisson Distribution 104
Exercises 108
5 Random Variables: The Continuous Case 119
5.1 Introduction/Purpose of the Chapter 119
5.2 Vignette/Historical Notes 119
5.3 Theory and Applications 120
5.3.1 Probability Density Function (p.d.f.) 120
5.3.2 Cumulative Distribution Function (c.d.f.) 124
5.3.3 Moments 127
5.3.4 Distribution of a Function of the Random Variable 128
5.4 Examples 130
5.4.1 Uniform Distribution on an Interval [ab] 130
5.4.2 Exponential Distribution 133
5.4.3 Normal Distribution (my, sigma²) 136
5.4.4 Gamma Distribution 139
5.4.5 Beta Distribution 144
5.4.6 Student's t Distribution 147
5.4.7 Pareto Distribution 149
5.4.8 The Log-Normal Distribution 151
5.4.9 Laplace Distribution 153
5.4.10 Double Exponential Distribution 155
Exercises 156
6 Generating Random Variables 177
6.1 Introduction/Purpose of the Chapter 177
6.2 Vignette/Historical Notes 178
6.3 Theory and Applications 178
6.3.1 Generating One-Dimensional Random Variables by Inverting the Cumulative Distribution Function (c.d.f.) 178
6.3.2 Generating One-Dimensional Normal Random Variables 183
6.3.3 Generating Random Variables. Rejection Sampling Method 186
6.3.4 Generating from a Mixture of Distributions 193
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"On the whole, the book has two features that set it apart from similar books: the full solutions and the examples from finance. It is up to you to decide if that makes it worth your time checking it out." ( Mathematical Association of America , 1 November 2014)
