A course in mathematical statistics

593 Pages·1997·2.377 MB·English

by  George G. Roussas

Contents i
A Course in Mathematical Statistics
Second Edition
ii Contents
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Contents iii
A Course in Mathematical Statistics
Second Edition
George G. Roussas
Intercollege Division of Statistics
University of California
Davis, California
ACADEMIC PRESS
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iv Contents
This book is printed on acid-free paper. ∞
Copyright © 1997 by Academic Press
All rights reserved.
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Library of Congress Cataloging-in-Publication Data
Roussas, George G.
A course in mathematical statistics / George G. Roussas.—2nd ed.
p. cm.
Rev. ed. of: A first course in mathematical statistics. 1973.
Includes index.
ISBN 0-12-599315-3
1. Mathematical statistics. I. Roussas, George G. First course in mathematical
statistics. II. Title.
QA276.R687 1997 96-42115
519.5—dc20 CIP
Printed in the United States of America
96 97 98 99 00 EB 9 8 7 6 5 4 3 2 1
Contents v
To my wife and sons
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Contents vii
Contents
Preface to the Second Edition xv
Preface to the First Edition xviii
Chapter 1 Basic Concepts of Set Theory 1
1.1 Some Definitions and Notation 1
Exercises 5
1.2* Fields and σσσσσ-Fields 8
Chapter 2 Some Probabilistic Concepts and Results 14
2.1 Probability Functions and Some Basic Properties and Results 14
Exercises 20
2.2 Conditional Probability 21
Exercises 25
2.3 Independence 27
Exercises 33
2.4 Combinatorial Results 34
Exercises 40
2.5* Product Probability Spaces 45
Exercises 47
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viii Contents
2.6* The Probability of Matchings 47
Exercises 52
Chapter 3 On Random Variables and Their Distributions 53
3.1 Some General Concepts 53
3.2 Discrete Random Variables (and Random Vectors) 55
Exercises 61
3.3 Continuous Random Variables (and Random Vectors) 65
Exercises 76
3.4 The Poisson Distribution as an Approximation to the Binomial
Distribution and the Binomial Distribution as an Approximation to the
Hypergeometric Distribution 79
Exercises 82
3.5* Random Variables as Measurable Functions and Related Results 82
Exercises 84
Chapter 4 Distribution Functions, Probability Densities, and Their
Relationship 85
4.1 The Cumulative Distribution Function (c.d.f. or d.f.) of a Random
Vector—Basic Properties of the d.f. of a Random Variable 85
Exercises 89
4.2 The d.f. of a Random Vector and Its Properties—Marginal and
Conditional d.f.’s and p.d.f.’s 91
Exercises 97
4.3 Quantiles and Modes of a Distribution 99
Exercises 102
4.4* Justification of Statements 1 and 2 102
Exercises 105
Chapter 5 Moments of Random Variables—Some Moment and
Probability Inequalities 106
5.1 Moments of Random Variables 106
Exercises 111
5.2 Expectations and Variances of Some R.V.’s 114
Exercises 119
5.3 Conditional Moments of Random Variables 122
Exercises 124
Contents ix
5.4 Some Important Applications: Probability and Moment
Inequalities 125
Exercises 128
5.5 Covariance, Correlation Coefficient and Its Interpretation 129
Exercises 133
5.6* Justification of Relation (2) in Chapter 2 134
Chapter 6 Characteristic Functions, Moment Generating Functions
and Related Theorems 138
6.1 Preliminaries 138
6.2 Definitions and Basic Theorems—The One-Dimensional Case 140
Exercises 145
6.3 The Characteristic Functions of Some Random Variables 146
Exercises 149
6.4 Definitions and Basic Theorems—The Multidimensional Case 150
Exercises 152
6.5 The Moment Generating Function and Factorial Moment Generating
Function of a Random Variable 153
Exercises 160
Chapter 7 Stochastic Independence with Some Applications 164
7.1 Stochastic Independence: Criteria of Independence 164
Exercises 168
7.2 Proof of Lemma 2 and Related Results 170
Exercises 172
7.3 Some Consequences of Independence 173
Exercises 176
7.4* Independence of Classes of Events and Related Results 177
Exercise 179
Chapter 8 Basic Limit Theorems 180
8.1 Some Modes of Convergence 180
Exercises 182
8.2 Relationships Among the Various Modes of Convergence 182
Exercises 187
8.3 The Central Limit Theorem 187
Exercises 194
8.4 Laws of Large Numbers 196
Exercises 198