An Introduction to Probability and Mathematical Statistics

An Introduction to Probability and Mathematical Statistics

1st Edition - January 1, 1962

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  • Author: Howard G. Tucker
  • eBook ISBN: 9781483225142

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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


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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