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Chapter 1 Events and Their Probabilities1

1.1 The History of Probability1

1.2 Experiment,Sample Space and Random Event3

1.2.1 Basic Definitions3

1.2.2 Events as Sets5

1.3 Probabilities Defined on Events8

1.3.1 Classical Probability8

1.3.2 Geometric Probability13

1.3.3 The Frequency Interpretation of Probability16

1.4 Probability Space18

1.4.1 Axiomatic Definition of Probability19

1.4.2 Properties of Probability20

1.5 Conditional Probabilities24

1.5.1 The Definition of Conditional Probability24

1.5.2 The Multiplication Rule28

1.5.3 Total Probability Formula30

1.5.4 Bayes'Theorem32

1.6 Independence of Events37

1.6.1 Independence of Two Events37

1.6.2 Independence of Several Events40

1.6.3 Bernoulli Trials44

1.7 Review45

1.8 Exercises46

Chapter 2 Random Variable54

2.1 The Definition of a Random Variable54

2.2 The Distribution Function of a Random Variable57

2.2.1 The Definition and Properties of Distribution Function57

2.2.2 The Distribution Function of Function of a Random Variable67

2.3 Mathematical Expectation and Variance71

2.3.1 Expectation of a Random Variable71

2.3.2 Expectation of Functions of a Random Variable77

2.3.3 Variance of a Random Variable80

2.3.4 The Application of Expectation and Variation85

2.4 Discrete Random Variables87

2.4.1 Binomial Distribution with Parameters n and p87

2.4.2 Geometric Distribution92

2.4.3 Poisson Distribution with Parametersλ95

2.5 Continuous Random Variables98

2.5.1 Uniform Distribution98

2.5.2 Exponential Distribution102

2.5.3 Normal Distribution107

2.6 Review114

2.7 Exercises117

Chapter 3 Random Vectors126

3.1 Random Vectors and Joint Distributions126

3.1.1 Random Vectors and Joint Distributions127

3.1.2 Discrete Random Vectors129

3.1.3 Continuous Random Vectors134

3.2 Independence of Random Variables141

3.3 Conditional Distributions148

3.3.1 Discrete Case148

3.3.2 Continuous Case150

3.4 One Function of Two Random Variables153

3.4.1 Discrete Case153

3.4.2 Continuous case157

3.5 Transformation of Two Random Variables164

3.6 Numerical Characteristics of Random Vectors167

3.6.1 Expectation of Sums and Products167

3.6.2 Covariance and Correlation170

3.7 Multivariate Distributions178

3.7.1 Distribution Functions of Multiple Random Vectors178

3.7.2 Numerical Characteristics of Random Vectors181

3.7.3 Multiple Normal Distribution186

3.8 Review188

3.9 Exercises191

Chapter 4 Sequences of Random Variables200

4.1 Family of Distribution Functions and Numerical Characteristics201

4.2 Modes of Convergence204

4.3 The Law of Large Numbers207

4.4 The Central Limit Theorem210

4.5 Review214

4.6 Exercises215

Chapter 5 Introduction to Stochastic Processes218

5.1 Definition and Classification218

5.2 The Distribution Family and the Moment Functions222

5.3 The Moments of the Stochastic Processes223

5.3.1 Mean,Autocorrelation and Autocovariance224

5.3.2 Cross-correlation and Cross-covariance227

5.4 Stochastic Analysis228

5.5 Review231

5.6 Exercises231

Chapter 6 Stationary Processes234

6.1 Stationary Processes234

6.1.1 Strict Stationary Processes234

6.1.2 Wide Stationary Processes236

6.1.3 Joint Stationary Processes241

6.2 Ergodicity of Stationary Processes242

6.3 Power Spectral Density of Stationary Processes246

6.3.1 Average Power and Power Spectral Density247

6.3.2 Power Spectral Density and Autocorrelation Function249

6.3.3 Cross-Power Spectral Density252

6.4 Stationary Processes and Linear Systems254

6.5 Review259

6.6 Exercises260

Chapter 7 Finite Markov Chains263

7.1 Basic Concepts263

7.2 Markov Chains Having Two States268

7.3 Higher Order Transition Probabilities and Distributions273

7.4 Invariant Distributions and Ergodic Markov Chain280

7.5 How Does Google Work？286

7.6 Review290

7.7 Exercises291

Chapter 8 Independent-Increment Processes297

8.1 Independent-Increment Processes297

8.2 Poisson Process298

8.3 Gaussian Processes305

8.4 Brownian Motion and Wiener Processes308

8.5 Review311

8.6 Exercises312

Bibliography316

Appendix318

Binom318

Table of Binomial Probabilities319

Table of Poisson Probabilities321

Table of Normal Probabilities324