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Probability Theory: An Elementary Course

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Chapter 1 Events and Probabilities
1．1 Random phenomena and statistical regularity
1．1．1 Random phenomena
1．1．2 The statistical definition of probability
1．2 Classical probability models
1．2．1 Sample points and sample spaces
1．2．2 Discrete probability models
1．2．3 Geometric probability models
1．3 The axiomatic definition of probability
1．3．1 Events
1．3．2 Probability space
1．3．3 Continuity of probability measure
1．4 Conditional probability and independent events
1．4．1 Conditional probability
1．4．2 Total probability formula and Bayes' rule
1．4．3 Independent events

Chapter 2 Random Variables and Distribution Functions
2．1 Discrete random variables
2．1．1 The concept of random variables
2．1．2 Discrete random variables
2．2 Distribution functions and continuous random variables
2．2．1 Distribution functions
2．2．2 Continuous random variables and density functions
2．2．3 Typical continuous random variables
2．3 Random vectors
2．3．1 Discrete random vectors
2．3．2 Joint distribution functions
2．3．3 Continuous random vectors
2．4 Independence of random variables
2．5 Conditional distribution
2．5．1 Discrete case
2．5．2 Continuous case
2．5．3 The general case
2．5．4 The conditional probability given a random variable
2．6 Functions of random variables
2．6．1 Functions of discrete random variables
2．6．2 Functions of continuous random variables
2．6．3 Functions of continuous random vectors
2．6．4 Transforms of random vectors
2．6．5 Important distributions in statistics

Chapter 3 Numerical Characteristics and Characteristic Functions
3．1 Mathematical expectations
3．1．1 Expectations of discrete random variables
3．1．2 Expectations of continuous random variables
3．1．3 General definition
3．1．4 Expectations of functions of random variables
3．1．5 Basic properties of expectations
3．1．6 Conditional expectation
3．2 Variances， covariances and correlation coefficients
3．2．1 Variances
3．2．2 Covariances
3．2．3 Correlation coefficients
3．2．4 Moments
3．3 Characteristic functions
3．3．1 Definitions
3．3．2 Properties
3．3．3 Inverse formula and uniqueness theorem
3．3．5 Multivariate characteristic functions
3．4 Multivariate normal distributions
3．4．1 Density functions and characteristic functions
3．4．2 Properties

Chapter 4 Probability Limit Theorems
4．1 Convergence in distribution and central limit theorems
4．1．1 Weak convergence of distribution functions
4．1．2 Central limit theorems
4．2 Convergence in probability and weak law of large numbers
4．2．1 Convergence in probability
4．2．2 Weak law of large numbers
4．3 Almost sure convergence and strong laws of large numbers
4．3．1 Almost sure convergence
4．3．2 Strong laws of large numbers
Bibliography

Appendix A Distribution of Typical Random Variables
A．1 Distribution of Typical Random Variables
A．2 Distributions of Typical Random Variables

Appendix B Tables
B．1 Table of Binomial Probabilities
B．2 Table of Random Digits
B．3 Table of Poisson Probabilities
B．4 Table of Standard Normal Distribution Function
B．5 Table of X2 Distribution
B．6 Table of t Distribution

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