What is included with this book?
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:
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.
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.
List of Figures xv
List of Tables xvii
Preface xix
Introduction xxi
1 Probability Space 1
1.1 Introduction/Purpose of the Chapter 1
1.2 Vignette/Historical Notes 2
1.3 Notations and Definitions 3
1.4 Theory and Applications 4
Problems 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
2.4 Examples 23
2.5 Monotone Convergence properties of probability 25
2.6 Conditional Probability 27
2.7 Independence of events and sigma fields 35
2.8 Borel Cantelli Lemmas 41
2.9 The Fatou lemmas 43
2.10 Kolmogorov zeroone law 44
2.11 Lebesgue measure on the unit interval (0,1] 45
Problems 48
3 Random Variables: Generalities 59
3.1 Introduction/ Purpose of the chapter 59
3.2 Vignette/Historical Notes 59
3.3 Theory and Applications 60
3.4 Independence of random variables 66
Problems 67
4 Random Variables: the discrete case 75
4.1 Introduction/Purpose of the chapter 75
4.2 Vignette/Historical Notes 76
4.3 Theory and Applications 76
4.4 Examples of discrete random variables 84
Problems 102
5 Random Variables: the continuous case 113
5.1 Introduction/purpose of the chapter 113
5.2 Vignette/Historical Notes 114
5.3 Theory and Applications 114
5.4 Moments 119
5.5 Change of variables 120
5.6 Examples 121
6 Generating Random variables 161
6.1 Introduction/Purpose of the chapter 161
6.2 Vignette/Historical Notes 162
6.3 Theory and applications 162
6.4 Generating multivariate distributions with prescribed covariance structure 188
Problems 191
7 Random vectors in Rn 193
7.1 Introduction/Purpose of the chapter 193
7.2 Vignette/Historical Notes 194
7.3 Theory and Applications 194
7.4 Distribution of sums of Random Variables. Convolutions 213
Problems 216
8 Characteristic Function 235
8.1 Introduction/Purpose of the chapter 235
8.2 Vignette/Historical Notes 235
8.3 Theory and Applications 236
8.4 The relationship between the characteristic function and the distribution 240
8.5 Examples 245
8.6 Gamma distribution 247
Problems 254
9 Momentgenerating function 259
9.1 Introduction/Purpose of the chapter 259
9.2 Vignette/ Historical Notes 260
9.3 Theory and Applications 260
Problems 272
10 Gaussian random vectors 277
10.1 Introduction/Purpose of the chapter 277
10.2 Vignette/Historical Notes 278
10.3 Theory and applications 278
Problems 300
11 Convergence Types. A.s. convergence. Lpconvergence. Convergence in probability. 313
11.1 Introduction/Purpose of the chapter 313
11.2 Vignette/Historical Notes 314
11.3 Theory and Applications: Types of Convergence 314
11.4 Relationships between types of convergence 320
Problems 333
12 Limit Theorems 345
12.1 Introduction/Purpose of the Chapter 345
12.2 Historical Notes 346
12.3 THEORY AND APPLICATIONS 348
12.4 Central Limit Theorem 372
Problems 380
Appendix A: Integration Theory. General Expectations 391
A.1 Integral of measurable functions 392
A.2 General Expectations and Moments of a Random Variable 399
Appendix B: Inequalities involving Random Variables and their Expectations 403
B.1 Functions of random variables. The Transport Formula. 409