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Chapter 1 Events and Their Probabilities 1

1.1 The History of Probability 1

1.2 Experiment,Sample Space and Random Event 3

1.2.1 Basic Definitions 3

1.2.2 Events as Sets 5

1.3 Probabilities Defined on Events 8

1.3.1 Classical Probability 8

1.3.2 Geometric Probability 13

1.3.3 The Frequency Interpretation of Probability 16

1.4 Probability Space 18

1.4.1 Axiomatic Definition of Probability 19

1.4.2 Properties of Probability 20

1.5 Conditional Probabilities 24

1.5.1 The Definition of Conditional Probability 24

1.5.2 The Multiplication Rule 28

1.5.3 Total Probability Formula 29

1.5.4 Bayes'Theorem 32

1.6 Independence of Events 36

1.6.1 Independence of Two Events 36

1.6.2 Independence of Several Events 40

1.6.3 Bernoulli Trials 43

1.7 Review 44

1.8 Exereises 45

Chapter 2 Random Variable 54

2.1 The Definition of a Random Variable 54

2.2 The Distribution Function of a Random Variable 56

2.2.1 The Definition and Properties of Distribution Function 57

2.2.2 The Distribution Function of Function of a Random Variable 67

2.3 Mathematical Expectation and Variance 71

2.3.1 Expectation of a Random Variable 71

2.3.2 Expectation of Functions of a Random Variable 77

2.3.3 Variance of a Random Variable 80

2.3.4 The Application of Expectation and Variation 85

2.4 Discrete Random Variables 87

2.4.1 Binomial Distribution with Parameters n and p 87

2.4.2 Geometric Distribution 92

2.4.3 Poisson Distribution with Parameters λ 95

2.5 Continuous Random Variables 98

2.5.1 Uniform Distribution 98

2.5.2 Exponential Distribution 102

2.5.3 Normal Distribution 107

2.6 Review 114

2.7 Exercises 117

Chapter 3 Random Vectors 126

3.1 Random Vectors and Joint Distributions 126

3.1.1 Random Vectors and Joint Distributions 127

3.1.2 Discrete Random Vectors 129

3.1.3 Continuous Random Vectors 134

3.2 Independence of Random Variables 141

3.3 Conditional Distributions 148

3.3.1 Discrete Case 148

3.3.2 Continuous Case 150

3.4 One Function of Two Random Variables 153

3.4.1 Discrete Case 153

3.4.2 Continuous case 157

3.5 Transformation of Two Random Variables 164

3.6 Numerical Characteristics of Random Vectors 167

3.6.1 Expectation of Sums and Products 167

3.6.2 Covariance and Correlation 171

3.7 Multivariate Distributions 178

3.7.1 Distribution Functions of Multiple Random Vectors 178

3.7.2 Numerical Characteristics of Random Vectors 181

3.7.3 Multiple Normal Distribution 186

3.8 Review 188

3.9 Exercises 191

Chapter 4 Sequences of Random Variables 200

4.1 Family of Distribution Functions and Numerical Characteristics 201

4.2 Modes of Convergence 204

4.3 The Law of Large Numbers 207

4.4 The Central Limit Theorem 210

4.5 Review 213

4.6 Exercises 214

Chapter 5 Introduction to Stochastic Processes 218

5.1 Definition and Classification 218

5.2 The Distribution Family and the Moment Functions 222

5.3 The Moments of the Stochastic Processes 223

5.3.1 Mean.Autocorrelation and Autocovariance 223

5.3.2 Cross-correlation and Cross-covariance 227

5.4 Stochastic Analysis 228

5.5 Review 231

5.6 Exercises 231

Chapter 6 Stationary Processes 233

6.1 Stationary Processes 233

6.1.1 Strict Stationary Processes 233

6.1.2 Wide Stationary Processes 235

6.1.3 Joint Stationary Processes 239

6.2 Ergodicity of Stationary Processes 241

6.3 Power Spectral Density of Stationary Processes 245

6.3.1 Average Power and Power Spectral Density 245

6.3.2 Power Spectral Density and Autocorrelation Function 248

6.3.3 Cross-Power Spectral Density 251

6.4 Stationary Processes and Linear Systems 253

6.5 Review 258

6.6 Exercises 259

Chapter 7 Finite Markov Chains 262

7.1 Basic Concepts 262

7.2 Markov Chains Having Two States 267

7.3 Higher Order Transition Probabilities and Distributions 272

7.4 Invariant Distributions and Ergodic Markov Chain 279

7.5 How Does Google Work？ 285

7.6 Review 289

7.7 Exercises 290

Chapter 8 Independent-Increment Processes 296

8.1 Independent-Increment Processes 296

8.2 Poisson Process 297

8.3 Gaussian Processes 304

8.4 Brownian Motion and Wiener Processes 307

8.5 Review 310

8.6 Exercises 311

Bibliography 315

Appendix 317

Table of Binomial Cofficients 317

Table of Binomial Probabilities 318

Table of Poisson Probabilities 320

Table of Normal Probabilities 323