Cover -- Title Page -- Copyright -- Contents in Brief -- Contents -- List of Figures -- Preface -- Introduction -- Chapter One: Probability Space -- 1.1 Introduction/Purpose of the Chapter -- 1.2 Vignette/Historical Notes -- 1.3 Notations and Definitions -- 1.4 Theory and Applications -- 1.4.1 Algebras -- 1.4.2 Sigma Algebras -- 1.4.3 Measurable Spaces -- 1.4.4 Examples -- 1.4.5 The Borel σ-algebra -- 1.5 Summary -- Exercises -- Chapter Two: Probability Measure -- 2.1 Introduction/Purpose of the Chapter -- 2.2 Vignette/Historical Notes -- 2.3 Theory and Applications -- 2.3.1 Definition and Basic Properties -- 2.3.2 Uniqueness of Probability Measures -- 2.3.3 Monotone Class -- 2.3.4 Examples -- 2.3.5 Monotone Convergence Properties of Probability -- 2.3.6 Conditional Probability -- 2.3.7 Independence of Events and σ-fields -- 2.3.8 Borel-Cantelli Lemmas -- 2.3.9 Fatou's Lemmas -- 2.3.10 Kolmogorov's Zero-One Law -- 2.4 Lebesgue Measure on the Unit Interval (0,1] -- Exercises -- Problems with Solution -- Problems Without Solution -- Chapter Three: Random Variables: Generalities -- 3.1 Introduction/Purpose of the Chapter -- 3.2 Vignette/Historical Notes -- 3.3 Theory and Applications -- 3.3.1 Definition -- 3.3.2 the Distribution of a Random Variable -- 3.3.3 the Cumulative Distribution Function of a Random Variable -- 3.3.4 Independence of Random Variables -- Exercises -- Problems with Solution -- Problems Without Solution -- Chapter Four: Random Variables: the Discrete Case -- 4.1 Introduction/Purpose of the Chapter -- 4.2 Vignette/Historical Notes -- 4.3 Theory and Applications -- 4.3.1 Definition and Basic Facts -- 4.3.2 Moments -- 4.4 Examples of Discrete Random Variables -- 4.4.1 The (discrete) Uniform Distribution.

4.4.2 Bernoulli Distribution -- 4.4.3 Binomial (n, p) Distribution -- 4.4.4 Geometric (p) Distribution -- 4.4.5 Negative Binomial (r, p) Distribution -- 4.4.6 Hypergeometric Distribution (N, m, n) -- 4.4.7 Poisson Distribution -- Exercises -- Problems with Solution -- Problems Without Solution -- Chapter Five: Random Variables: the Continuous Case -- 5.1 Introduction/Purpose of the Chapter -- 5.2 Vignette/Historical Notes -- 5.3 Theory and Applications -- 5.3.1 Probability Density Function (p.d.f.) -- 5.3.2 Cumulative Distribution Function (c.d.f.) -- 5.3.3 Moments -- 5.3.4 Distribution of a Function of the Random variable -- 5.4 Examples -- 5.4.1 Uniform Distribution on an Interval [a,b] -- 5.4.2 Exponential Distribution -- 5.4.3 Normal Distribution (µ, σ2) -- 5.4.4 Gamma Distribution -- 5.4.5 Beta Distribution -- 5.4.6 Student's t Distribution -- 5.4.7 Pareto Distribution -- 5.4.8 The Log-normal Distribution -- 5.4.9 Laplace Distribution -- 5.4.10 Double Exponential Distribution -- Exercises -- Problems with Solution -- Problems Without Solution -- Chapter Six: Generating Random Variables -- 6.1 Introduction/Purpose of the Chapter -- 6.2 Vignette/Historical Notes -- 6.3 Theory and Applications -- 6.3.1 Generating One-dimensional Random Variables by Inverting the Cumulative Distribution Function (c.d.f.) -- 6.3.2 Generating One-dimensional Normal Random Variables -- 6.3.3 Generating Random Variables. Rejection Sampling Method -- 6.3.4 Generating from a Mixture of Distributions -- 6.3.5 Generating Random Variables. Importance Sampling -- 6.3.6 Applying Importance Sampling -- 6.3.7 Practical Consideration: Normalizing Distributions -- 6.3.8 Sampling Importance Resampling -- 6.3.9 Adaptive Importance Sampling.

6.4 Generating Multivariate Distributions with Prescribed Covariance Structure -- Exercises -- Chapter Seven: Random Vectors in Rn -- 7.1 Introduction/Purpose of the Chapter -- 7.2 Vignette/Historical Notes -- 7.3 Theory and Applications -- 7.3.1 The Basics -- 7.3.2 Marginal Distributions -- 7.3.3 Discrete Random Vectors -- 7.3.4 Multinomial Distribution -- 7.3.5 Testing Whether Counts are Coming from a Specific Multinomial Distribution -- 7.3.6 Independence -- 7.3.7 Continuous Random Vectors -- 7.3.8 Change of Variables. Obtaining Densities of Functions of Random Vectors -- 7.3.9 Distribution of Sums of Random Variables. Convolutions -- Exercises -- Problems with Solution -- Problems Without Solution -- Chapter Eight: Characteristic Function -- 8.1 Introduction/Purpose of the Chapter -- 8.2 Vignette/Historical Notes -- 8.3 Theory and Applications -- 8.3.1 Definition and Basic Properties -- 8.3.2 the Relationship Between the Characteristic Function and the Distribution -- 8.4 Calculation of the Characteristic Function for Commonly Encountered Distributions -- 8.4.1 Bernoulli and Binomial -- 8.4.2 Uniform Distribution -- 8.4.3 Normal Distribution -- 8.4.4 Poisson Distribution -- 8.4.5 Gamma Distribution -- 8.4.6 Cauchy Distribution -- 8.4.7 Laplace Distribution -- 8.4.8 Stable Distributions. Lévy Distribution -- 8.4.9 Truncated Lévy Flight Distribution -- Exercises -- Problems with Solution -- Problems Without Solution -- Chapter Nine: Moment-generating Function -- 9.1 Introduction/Purpose of the Chapter -- 9.2 Vignette/Historical Notes -- 9.3 Theory and Applications -- 9.3.1 Generating Functions and Applications.

9.3.2 Moment-generating Functions. Relation with the Characteristic Functions -- 9.3.3 Relationship with the Characteristic Function -- 9.3.4 Properties of the MGF -- Exercises -- Problems with Solution -- Problems Without Solution -- Chapter Ten: Gaussian Random Vectors -- 10.1 Introduction/Purpose of the Chapter -- 10.2 Vignette/Historical Notes -- 10.3 Theory and Applications -- 10.3.1 The Basics -- 10.3.2 Equivalent Definitions of a Gaussian Vector -- 10.3.3 Uncorrelated Components and Independence -- 10.3.4 The Density of a Gaussian Vector -- 10.3.5 Cochran's Theorem -- 10.3.6 Matrix Diagonalization and Gaussian Vectors -- Exercises -- Problems with Solution -- Problems Without Solution -- Chapter Eleven: Convergence Types. Almost Sure Convergence. Lp-convergence. Convergence in Probability -- 11.1 Introduction/Purpose of the Chapter -- 11.2 Vignette/Historical Notes -- 11.3 Theory and Applications: Types of Convergence -- 11.3.1 Traditional Deterministic Convergence Types -- 11.3.2 Convergence of Moments of an r.v.-convergence in Lp -- 11.3.3 Almost Sure (a.s.) Convergence -- 11.3.4 Convergence in Probability -- 11.4 Relationships Between Types of Convergence -- 11.4.1 a.s. and Lp -- 11.4.2 Probability and a.s./lp -- 11.4.3 Uniform Integrability -- Exercises -- Problems with Solution -- Problems Without Solution -- Chapter Twelve: Limit Theorems -- 12.1 Introduction/Purpose of the Chapter -- 12.2 Vignette/Historical Notes -- 12.3 Theory and Applications -- 12.3.1 Weak Convergence -- 12.3.2 The Law of Large Numbers -- 12.4 Central Limit Theorem -- Exercises -- Problems with Solution -- Problems Without Solution.

Chapter Thirteen: Appendix A: Integration Theory. General Expectations -- 13.1 Integral of Measurable Functions -- 13.1.1 Integral of Simple (elementary) Functions -- 13.1.2 Integral of Positive Measurable Functions -- 13.1.3 Integral of Measurable Functions -- 13.2 General Expectations and Moments of a Random Variable -- 13.2.1 Moments and Central Moments. Lp Space -- 13.2.2 Variance and the Correlation Coefficient -- 13.2.3 Convergence Theorems -- Chapter Fourteen: Appendix B: Inequalities Involving Random Variables and Their Expectations -- 14.1 Functions of Random Variables. the Transport Formula -- Bibliography -- Index.