Chapter 1Events and Probabilities
1.1Random phenomena and statistical regularity
1.1.1Random phenomena
1.1.2The statistical definition of probability
1.2Classical probability models
1.2.1Sample points and sample spaces
1.2.2Discrete probability models
1.2.3Geometric probability models
1.3The axiomatic definition of probability
1.3.1Events
1.3.2Probability space
1.3.3Continuity of probability measure
1.4Conditional probability and independent events
1.4.1Conditional probability
1.4.2Total probability formula and Bayes'' rule
1.4.3Independent events
Chapter 2Random Variables and Distribution Functions
2.1Discrete random variables
2.1.1The concept of random variables
2.1.2Discrete random variables
2.2Distribution functions and continuous random variables
2.2.1Distribution functions
2.2.2Continuous random variables and density functions
2.2.3Typical continuous random variables
2.3Random vectors
2.3.1Discrete random vectors
2.3.2Joint distribution functions
2.3.3Continuous random vectors
2.4Independence of random variables
2.5Conditional distribution
2.5.1Discrete case
2.5.2Continuous case
2.5.3The general case
2.5.4The conditional probability given a random variable
2.6Functions of random variables
2.6.1Functions of discrete random variables
2.6.2Functions of continuous random variables
2.6.3Functions of continuous random vectors
2.6.4Transforms of random vectors
2.6.5Important distributions in statistics
Chapter 3Numerical Characteristics and Characteristic Functions
3.1Mathematical expectations
3.1.1Expectations of discrete random variables
3.1.2Expectations of continuous random variables
3.1.3General definition
3.1.4Expectations of functions of random variables
3.1.5Basic properties of expectations
3.1.6Conditional expectation
3.2Variances, covariances and correlation coefficients
3.2.1Variances
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