An Introduction to Probability and Mathematical Statistics
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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.
Table of Contents
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
Product details
- No. of pages: 240
- Language: English
- Copyright: © Academic Press 1962
- Published: January 1, 1962
- Imprint: Academic Press
- eBook ISBN: 9781483225142
About the Author
Howard G. Tucker
About the Editor
Ralph P. Boas
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