Probability Theory:An Elementary Course

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£®4 Additivity of distribution functions
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