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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 30

1.5.4 Bayes'Theorem 32

1.6 Independence of Events 37

1.6.1 Independence of Two Events 37

1.6.2 Independence of Several Events 40

1.6.3 Bernoulli Trials 44

1.7 Review 45

1.8 Exercises 46

Chapter 2 Random Variable 54

2.1 The Definition of a Random Variable 54

2.2 The Distribution Function of a Random Variable 57

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 Parametersλ 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 170

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 214

4.6 Exercises 215

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 224

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 234

6.1 Stationary Processes 234

6.1.1 Strict Stationary Processes 234

6.1.2 Wide Stationary Processes 236

6.1.3 Joint Stationary Processes 241

6.2 Ergodicity of Stationary Processes 242

6.3 Power Spectral Density of Stationary Processes 246

6.3.1 Average Power and Power Spectral Density 247

6.3.2 Power Spectral Density and Autocorrelation Function 249

6.3.3 Cross-Power Spectral Density 252

6.4 Stationary Processes and Linear Systems 254

6.5 Review 259

6.6 Exercises 260

Chapter 7 Finite Markov Chains 263

7.1 Basic Concepts 263

7.2 Markov Chains Having Two States 268

7.3 Higher Order Transition Probabilities and Distributions 273

7.4 Invariant Distributions and Ergodic Markov Chain 280

7.5 How Does Google Work？ 286

7.6 Review 290

7.7 Exercises 291

Chapter 8 Independent-Increment Processes 297

8.1 Independent-Increment Processes 297

8.2 Poisson Process 298

8.3 Gaussian Processes 305

8.4 Brownian Motion and Wiener Processes 308

8.5 Review 311

8.6 Exercises 312

Bibliography 316

Appendix 318

Binom 318

Table of Binomial Probabilities 319

Table of Poisson Probabilities 321

Table of Normal Probabilities 324