Tucker / Boas An Introduction to Probability and Mathematical Statistics

1. Auflage 2014
ISBN: 978-1-4832-2514-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 240 Seiten, Web PDF

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.

Autoren/Hrsg.

Tucker, Howard G.

Boas, Ralph P.

Inhaltsverzeichnis

1;Front Cover;1
2;An Introduction to Probability and Mathematical Statistics;4
3;Copyright Page;5
4;Table of Contents;12
5;Dedication;6
6;Preface;8
7;Chapter 1. Events and Probabilities;14
7.1;1.1 Combinatorial Probability;14
7.2;1.2 The Fundamental Probability Set and the Algebra of Events;19
7.3;1.3 The Axioms of a Probability Space;22
8;Chapter 2. Dependent and Independent Events;27
8.1;2.1 Conditional Probability;27
8.2;2.2 Stochastic Independence;31
8.3;2.3 An Application in Physics of the Notion of Independence;34
9;Chapter 3. Random Variables and Probability Distributions;41
9.1;3.1 The Definition of a Function;41
9.2;3.2 The Definition of a Random Variable;43
9.3;3.3 Combinations of Random Variables;46
9.4;3.4 Distribution Functions;49
9.5;3.5 Multivariate Distribution Functions;53
10;Chapter 4. Discrete Distributions;57
10.1;4.1 Univariate Discrete Distributions;57
10.2;4.2 The Binomial and Pascal Distributions;58
10.3;4.3 The Hypergeometric Distribution;61
10.4;4.4 The Poisson Distribution;62
10.5;4.5 Multivariate Discrete Densities;63
11;Chapter 5. Absolutely Continuous Distributions;67
11.1;5.1 Absolutely Continuous Distributions;67
11.2;5.2 Densities of Functions of Random Variables;72
12;Chapter 6. Some Special Absolutely Continuous Distributions;79
12.1;6.1 The Gamma and Beta Functions;79
12.2;6.2 The Normal Distribution;82
12.3;6.3 The Negative Exponential Distribution;84
12.4;6.4 The Chi-Square Distribution;86
12.5;6.5 The F-Distribution and the t-Distribution;89
13;Chapter 7. Expectation and Limit Theorems;94
13.1;7.1 Definition of Expectation;94
13.2;7.2 Expectation of Functions of Random Variables;99
13.3;7.3 Moments and Central Moments;103
13.4;7.4 Convergence in Probability;108
13.5;7.5 Limit Theorems;111
14;Chapter 8. Point Estimation;117
14.1;8.1 Sampling;117
14.2;8.2 Unbiased and Consistent Estimates;122
14.3;8.3 The Method of Moments;126
14.4;8.4 Minimum Variance Estimates;128
14.5;8.5 The Principle of Maximum Likelihood;134
15;Chapter 9. Notes on Matrix Theory;138
16;Chapter 10. The Multivariate Normal Distribution;159
16.1;10.1 The Multivariate Normal Density;159
16.2;10.2 Properties of the Multivariate Normal Distribution;162
16.3;10.3 Cochran's Theorem;167
16.4;10.4 Proof of the Independence of the Sample Mean and Sample Variance for a Normal Population;171
17;Chapter 11. Testing Statistical Hypotheses: Simple Hypothesis vs. Simple Alternative;174
17.1;11.1 Fundamental Notions of Hypothesis Testing;174
17.2;11.2 Simple Hypothesis vs. Simple Alternative;177
17.3;11.3 The Neyman-Pearson Fundamental Lemma;181
17.4;11.4 Randomized Tests;188
18;Chapter 12. Testing Simple and Composite Hypotheses;196
18.1;12.1 Uniformly Most Powerful Critical Regions;196
18.2;12.2 The Likelihood Ratio Test;203
18.3;12.3 The t-Test;211
18.4;12.4 The Analysis of Variance;213
19;Chapter 13. Confidence Intervals;220
19.1;13.1 The Neyman Theory of Confidence Intervals;220
19.2;13.2 The Relation between Confidence Intervals and Tests of Hypotheses;224
19.3;13.3 Necessary and Sufficient Conditions for the Existence of Confidence Intervals;229
20;Suggested Reading;232
21;Tables I—IV;233
22;Index;238