Probability and Mathematical Statistics

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.

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