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
