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 R¯n

* 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.

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.

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.

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

3.3.2 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 (_ _2) 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

6.3.5 Generating Random Variables. Importance Sampling 195

6.3.6 Applying Importance Sampling 198

6.3.7 Practical Consideration: Normalizing Distributions 201

6.3.8 Sampling Importance Resampling 203

6.3.9 Adaptive Importance Sampling 204

6.4 Generating Multivariate Distributions with Prescribed Covariance Structure 205

Exercises 208

7 Random Vectors in Rn 210

7.1 Introduction/Purpose of the Chapter 210

7.2 Vignette/Historical Notes 210

7.3 Theory and Applications 211

7.3.1 The Basics 211

7.3.2 Marginal Distributions 212

7.3.3 Discrete Random Vectors 214

7.3.4 Multinomial Distribution 219

7.3.5 Testing Whether Counts are Coming from a Specific Multinom…