Probability and Mathematical Statistics: An Introduction provides a well-balanced first introduction to probability theory and mathematical statistics.
This book is organized into two sections encompassing nine chapters. The first part deals with the concept and elementary properties of probability space, and random variables and their probability distributions. This part also considers the principles of limit theorems, the distribution of random variables, and the so-called student's distribution. The second part explores pertinent topics in mathematical statistics, including the concept of sampling, estimation, and hypotheses testing.
This book is intended primarily for undergraduate statistics students.
Inhalt
Preface
Introduction
Part I. Probability Theory
Chapter 1. The Probability Space
1.1 The Outcome Space
1.2 Probabilities
1.3 The Axioms
1.4 Problems
References
Chapter 2. Elementary Properties of Probability Spaces
2.1 Simple Consequences of the Axioms
2.2 Conditional Probability and Independence
2.3 Finite Probability Spaces
2.4 Problems
Chapter 3. Random Variables and Their Probability Distributions
3.1 Random Variables
3.2 Distribution Functions
3.3 Examples of Discrete Distributions
3.4 Examples of Absolutely Continuous Distributions
3.5 Multivariate Distributions
3.6 Problems
References
Chapter 4. Typical Values
4.1 The Mathematical Expectation of a Random Variable
4.2 Expectations of Functions of Random Variables
4.3 Properties of Expectations
4.4 Moments
4.5 Regression
4.6 Problems
Reference
Chapter 5. Limit Theorems
5.1 Laws of Large Numbers
5.2 The Central Limit Theorem
5.3 The Poisson Approximation to the Binomial
5.4 Problems
References
Chapter 6. Some Important Distributions
6.1 The Distribution of the Sum of Independent, Absolutely Continuous Random Variables
6.2 Addition of Independent Normal Random Variables
6.3 The Chi-Square Distribution
6.4 Student's Distribution
6.5 Problems
Part II. Mathematical Statistics
Chapter 7. Sampling
7.1 Statistical Data
7.2 Sample Characteristics
7.3 Moments and Distributions of Sample Characteristics
7.4 Problems
References
Chapter 8. Estimation
8.1 Properties of Estimates
8.2 Point Estimation
8.3 Interval Estimation
8.4 Problems
References
Chapter 9. Testing Hypotheses
9.1 Statistical Hypotheses
9.2 The Power of a Test
9.3 The t-Test
9.4 Nonparametric Methods
9.5 Problems
References
Appendix A. Some Combinatorial Formulas
Appendix B. The Gamma Function
Appendix C. Proof of the Central Limit Theorem
Appendix D. Tables
Answers to Selected Problems
Index