概率论讲义（英文版）

Contents
丛书序
序言
Preface
Chapter 1 Probability and Probability Space
1.1 What is Probability?
1.2 Sample Space and Events
1.3 Definition of Probability
1.3.1 Classic Probability
1.3.2 Empirical Probability
1.3.3 Geometrical Probability
1.4 Axioms of Probability and Probability Space
1.4.1 Algebra and σ-Algebra
1.4.2 Axioms of Probability
1.5 Conditional Probability
1.5.1 Definition of Conditional Probability
1.5.2 Law of Total Probability and Bayes5 Formula
1.5.3 Independent Events
Chapter 2 Random Variables and Distribution Functions
2.1 The Distribution Function of a Random Variable
2.2 Discrete Random Variables
2.2.1 Definition of a Discrete Random Variable
2.2.2 The Bernoulli Random Variable
2.2.3 The Poisson Random Variable
2.3 Continuous Random Variables
2.3.1 Definition of a Continuous Random Variable
2.3.2 Normal Random Variable
2.3.3 Other Continuous Random Variables
Chapter 3 Jointly Distributed Random Variables
3.1 The Joint Distribution Function
3.1.1 Jointly Distributed Discrete Random Variables
3.1.2 Jointly Distributed Continuous Random Variables
3.1.3 The Marginal Distribution
3.2 Independent Random Variables
3.3 The Conditional Distribution
3.3.1 The Jointly Distributed Discrete Random Variables Case
3.3.2 The Jointly Distributed Continuous Random Variables Case
3.4 The Joint Probability Distribution of Functions of Random Variables
3.4.1 Key Theorem
3.4.2 Transformations of Two Random Variables
Chapter 4 Expectation and Variance of Random Variables
4.1 Expectation and Variance of a Discrete Random Variable
4.2 Expectation and Variance of a Continuous Random Variable
4.3 General Definition of Expectation
4.4 Moments of a Random Variable
4.5 Geometric Property of Expectation
4.6 Expectation of Jointly Distributed Random Variables
4.6.1 Two Dimensional Riemann-Stieltjes Integral
4.6.2 Covariance of Jointly Distributed Random Variables
4.6.3 Expectation of Functions of Jointly Distributed Random Variables
4.6.4 Correlation
Chapter 5 Characteristic Functions of Random Variables
5.1 The Characteristic Function of a Random Variable
5.2 The Inversion Formula of the Characteristic Function
5.3 The Joint Characteristic Function
Chapter 6 Large Number Laws and Central Limit Theorem
6.1 Convergence in Probability Theory
6.2 Laws of Large Numbers
6.2.1 Weak Law of Large Numbers
6.2.2 Strong Law of Large Numbers
6.3 Central Limit Theorem
6.3.1 The Central Limit Theorem
6.3.2 Linderberg-Feller Theorem
6.4 Proofs of Theorems 5.1.2, 6.2.3 and 6.3.
References
Appendix 1 Numerical Table for Poisson Distribution
Appendix 2 Numerical Table for Standard Normal Distribution
Appendix 3 Translation of Some Mathematical Professional Terms (部分专业词汇对照表）
Appendix 4 Translation of Some Mathematicians (译名对照表）
Index