Contents
丛书序
序言
Preface
Chapter 1 Probability and Probability Space 1
1.1 What is Probability? 1
1.2 Sample Space and Events 2
1.3 Definition of Probability 4
1.3.1 Classic Probability 4
1.3.2 Empirical Probability 6
1.3.3 Geometrical Probability 7
1.4 Axioms of Probability and Probability Space 10
1.4.1 Algebra and -Algebra 10
1.4.2 Axioms of Probability 10
1.5 Conditional Probability 14
1.5.1 Definition of Conditional Probability 14
1.5.2 Law of Total Probability and Bayes5 Formula 16
1.5.3 Independent Events 18
Chapter 2 Random Variables and Distribution Functions 21
2.1 The Distribution Function of a Random Variable 21
2.2 Discrete Random Variables 24
2.2.1 Definition of a Discrete Random Variable 24
2.2.2 The Bernoulli Random Variable 25
2.2.3 The Poisson Random Variable 26
2.3 Continuous Random Variables 33
2.3.1 Definition of a Continuous Random Variable 33
2.3.2 Normal Random Variable 34
2.3.3 Other Continuous Random Variables 36
Chapter 3 Jointly Distributed Random Variables 40
3.1 The Joint Distribution Function 40
3.1.1 Jointly Distributed Discrete Random Variables 41
3.1.2 Jointly Distributed Continuous Random Variables 41
3.1.3 The Marginal Distribution 42
3.2 Independent Random Variables 44
3.3 The Conditional Distribution 46
3.3.1 The Jointly Distributed Discrete Random Variables Case 46
3.3.2 The Jointly Distributed Continuous Random Variables Case 47
3.4 The Joint Probability Distribution of Functions of Random Variables 48
3.4.1 Key Theorem 49
3.4.2 Transformations of Two Random Variables 50
Chapter 4 Expectation and Variance of Random Variables 56
4.1 Expectation and Variance of a Discrete Random Variable 56
4.2 Expectation and Variance of a Continuous Random Variable 58
4.3 General Definition of Expectation 61
4.4 Moments of a Random Variable 66
4.5 Geometric Property of Expectation 68
4.6 Expectation of Jointly Distributed Random Variables 69
4.6.1 Two Dimensional Riemann-Stieltjes Integral 69
4.6.2 Covariance of Jointly Distributed Random Variables 71
4.6.3 Expectation of Functions of Jointly Distributed Random Variables 73
4.6.4 Correlation 76
Chapter 5 Characteristic Functions of Random Variables 79
5.1 The Characteristic Function of a Random Variable 79
5.2 The Inversion Formula of the Characteristic Function 84
5.3 The Joint Characteristic Function 88
Chapter 6 Large Number Laws and Central Limit Theorem 93
6.1 Convergence in Probability Theory 93
6.2 Laws of Large Numbers 97
6.2.1 Weak Law of Large Numbers 97
6.2.2 Strong Law of Large Numbers 99
6.3 Central Limit Theorem 101
6.3.1 The Central Limit Theorem 101
6.3.2 Linderberg-Feller Theorem 105
6.4 Proofs of Theorems 5.1.2， 6.2.3 and 6.3.2 112
References 126
Appendix 1 Numerical Table for Poisson Distribution 127
Appendix 2 Numerical Table for Standard Normal Distribution 133
Appendix 3 Translation of Some Mathematical Professional Terms (部分专业词汇对照表） 135
Appendix 4 Translation of Some Mathematicians (译名对照表） 137
Index 138
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