An Introduction to Probability and Mathematical Statistics

An Introduction to Probability and Mathematical Statistics provides information pertinent to the fundamental aspects of probability and mathematical statistics. This book covers a variety of topics, including random variables, probability distributions, discrete distributions, and point estimation.

Organized into 13 chapters, this book begins with an overview of the definition of function. This text then examines the notion of conditional or relative probability. Other chapters consider Cochran's theorem, which is of extreme importance in that part of statistical inference known as analysis of variance. This book discusses as well the fundamental principles of testing statistical hypotheses by providing the reader with an idea of the basic problem and its relation to practice. The final chapter deals with the problem of estimation and the Neyman theory of confidence intervals.

This book is a valuable resource for undergraduate university students who are majoring in mathematics. Students who are majoring in physics and who are inclined toward abstract mathematics will also find this book useful.

Preface

Chapter 1 Events and Probabilities

1.1 Combinatorial Probability

1.2 The Fundamental Probability Set and the Algebra of Events

1.3 The Axioms of a Probability Space

Chapter 2 Dependent and Independent Events

2.1 Conditional Probability

2.2 Stochastic Independence

2.3 An Application in Physics of the Notion of Independence

Chapter 3 Random Variables and Probability Distributions

3.1 The Definition of a Function

3.2 The Definition of a Random Variable

3.3 Combinations of Random Variables

3.4 Distribution Functions

3.5 Multivariate Distribution Functions

Chapter 4 Discrete Distributions

4.1 Univariate Discrete Distributions

4.2 The Binomial and Pascal Distributions

4.3 The Hypergeometric Distribution

4.4 The Poisson Distribution

4.5 Multivariate Discrete Densities

Chapter 5 Absolutely Continuous Distributions

5.1 Absolutely Continuous Distributions

5.2 Densities of Functions of Random Variables

Chapter 6 Some Special Absolutely Continuous Distributions

6.1 The Gamma and Beta Functions

6.2 The Normal Distribution

6.3 The Negative Exponential Distribution

6.4 The Chi-Square Distribution

6.5 The F-Distribution and the t-Distribution

Chapter 7 Expectation and Limit Theorems

7.1 Definition of Expectation

7.2 Expectation of Functions of Random Variables

7.3 Moments and Central Moments

7.4 Convergence in Probability

7.5 Limit Theorems

Chapter 8 Point Estimation

8.1 Sampling

8.2 Unbiased and Consistent Estimates

8.3 The Method of Moments

8.4 Minimum Variance Estimates

8.5 The Principle of Maximum Likelihood

Chapter 9 Notes on Matrix Theory

Chapter 10 The Multivariate Normal Distribution

10.1 The Multivariate Normal Density

10.2 Properties of the Multivariate Normal Distribution

10.3 Cochran's Theorem

10.4 Proof of the Independence of the Sample Mean and Sample Variance for a Normal Population

Chapter 11 Testing Statistical Hypotheses: Simple Hypothesis vs. Simple Alternative

11.1 Fundamental Notions of Hypothesis Testing

11.2 Simple Hypothesis vs. Simple Alternative

11.3 The Neyman-Pearson Fundamental Lemma

11.4 Randomized Tests

Chapter 12 Testing Simple and Composite Hypotheses

12.1 Uniformly Most Powerful Critical Regions

12.2 The Likelihood Ratio Test

12.3 The t-Test

12.4 The Analysis of Variance

Chapter 13 Confidence Intervals

13.1 The Neyman Theory of Confidence Intervals

13.2 The Relation Between Confidence Intervals and Tests of Hypotheses

13.3 Necessary and Sufficient Conditions for the Existence of Confidence Intervals

Suggested Reading

Tables I—IV

Index