Chapter 1 Probability and Its Properties
1.1 Basic Probability Concepts
1.2 Axioms and Properties of Probability
1.2.1 Axioms Definition of Probability
1.2.2 Properties of Probability
1.3 Classical Probability and Counting Techniques
1.3.1 Counting Principles
1.3.2 Classical Probability
1.4 Conditional Probability, Independence of Two and Several Events
1.4.1 Conditional Probability
1.4.2 Independence
1.5 Law of Total Probability and Bayes' Theorem
Exercises
Chapter 2 Random Variables and Their Distributions
2.1 Random Variables
2.2 Distribution of a Random Variable and Distribution Function
2.3 Classical Discrete Random Variables and Continuous Random Variables
2.3.1 Discrete Distribution
2.3.2 Continuous Distribution
2.4 Distribution of Functions of a Random Variable
Exercises
Chapter 3 Random Vectors and Their Distributions
3.1 Jointly Distributed Random Variables
3.2 Marginal Distribution and Conditional Distribution of Two Random Variables
3.3 Independent Random Variables
3.4 Distribution of Functions of Two Random Variables
Exercises
Chapter4 Expectations and Moments
4.1 Mathematical Expectation and Its Properties
4.1.1 Mathematical Expectation
4.1.2 Properties of the Expectation
4.2 Variance and Its Properties
4.2.1 Definition of the Variance
4.2.2 Properties of the Variance
4.3 Expectations and Variances of Special Probability Distributions
4.3.1 Case for Common Discrete Random Variables
4.3.2 Case for Common Continuous Random Variables
4.4 Moments
4.4.1 Covariance and Correlation Coefficients
4.4.2 Moments
Exercises
Chapter 5 The Law of Large Numbers and the Central Limit Theorem
5. I The Law of Large Numbers and Its Applications
5.1.1 Chebyshev's Inequality
5.1.2 The Law of Large Numbers
5.2 The Central Limit Theorem and Its Applications
Exercises
Chapter 6 Basic Conceptions of Statistics
6.1 Basic Conceptions of Sampling
6.2 Descriptive Statistics
6.2.1 Summarizing Data--Numerical Methods
6.2.2 Summarizing Data--Graphical Methods
6.3 Fundamental Sampling Distributions
6.3.1 The Chi-squared Distribution
6.3.2 The t-Distribution
6.3.3 The F-Distribution
6.4 Sampling Distribution Theorems
Exercises
Chapter 7 Parameter Estimation
7.1 General Concepts of Point Estimation
7.2 Methods of Point Estimation
7.2.1 Method of Moments
7.2.2 Method of Maximum Likelihood Estimation
7.3 Criteria for Good Estimators
7.4 Interval Estimation
7.4.1 Confidence Intervals Based on a Single Sample
7.4.2 Confidence Intervals Based on Two Samples
7.5 One-sided Confidence Intervals (Confidence Bounds)
Exercises
Chapter 8 Hypothesis Testing
8.1 Hypotheses and Testing Procedures
8.1.1 Hypotheses Testing Terminology
8.1.2 Testing Procedures
8.2 Tests Concerning Means and Variances
8.2.1 Tests About One Population Mean
8.2.2 Testing One Population Variance
8.2.3 Comparing Two Population Means
8.2.4 Comparing Two Population Variances
8.3 Duality Between Confidence Interval and Hypothesis Testing
Exercises
Chapter 9 Understanding Monte Carlo Method and Statistics Software
9.1 Monte Carlo Method
9.1.1 The Monte Carlo Method
9.1.2 Bootstrap Procedures
9.2 Introduction of Statistics Software
Appendix A Tables
A.1 Standard Normal Curve Areas
A.2 Critical Values for Chi-squared Distributions
A.3 Critical Values for t-Distributions
A.4 Critical Values for F-Distributions
Bibliography
