An Introduction to Probability and Statistics - Vijay K. Rohatgi - ebook

An Introduction to Probability and Statistics ebook

Vijay K. Rohatgi

489,99 zł


A well-balanced introduction to probability theory and mathematical statistics Featuring updated material, An Introduction to Probability and Statistics, Third Edition remains a solid overview to probability theory and mathematical statistics. Divided intothree parts, the Third Edition begins by presenting the fundamentals and foundationsof probability. The second part addresses statistical inference, and the remainingchapters focus on special topics. An Introduction to Probability and Statistics, Third Edition includes: * A new section on regression analysis to include multiple regression, logistic regression, and Poisson regression * A reorganized chapter on large sample theory to emphasize the growing role of asymptotic statistics * Additional topical coverage on bootstrapping, estimation procedures, and resampling * Discussions on invariance, ancillary statistics, conjugate prior distributions, and invariant confidence intervals * Over 550 problems and answers to most problems, as well as 350 worked out examples and 200 remarks * Numerous figures to further illustrate examples and proofs throughout An Introduction to Probability and Statistics, Third Edition is an ideal reference and resource for scientists and engineers in the fields of statistics, mathematics, physics, industrial management, and engineering. The book is also an excellent text for upper-undergraduate and graduate-level students majoring in probability and statistics.

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Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Geof H. Givens, Harvey Goldstein, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford WeisbergEditors Emeriti: J. Stuart Hunter, Iain M. Johnstone, Joseph B. Kadane, Jozef L. Teugels

A complete list of the titles in this series appears at the end of this volume.


Third Edition













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Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Rohatgi, V. K., 1939-An introduction to probability theory and mathematical statistics / Vijay K. Rohatgi and A. K. Md. Ehsanes Saleh. - 3rd edition.      pages cm   Includes index.

   ISBN 978-1-118-79964-2 (cloth)1. Probabilities. 2. Mathematical statistics. I. Saleh, A. K. Md. Ehsanes. II. Title.   QA273.R56 2015   519.5-dc23





To Bina and Shahidara.


The Third Edition contains some new material. More specifically, the chapter on large sample theory has been reorganized, repositioned, and re-titled in recognition of the growing role of asymptotic statistics. In Chapter 12 on General Linear Hypothesis, the section on regression analysis has been greatly expanded to include multiple regression and logistic and Poisson regression.

Some more problems and remarks have been added to illustrate the material covered. The basic character of the book, however, remains the same as enunciated in the Preface to the first edition. It remains a solid introduction to first-year graduate students or advanced seniors in mathematics and statistics as well as a reference to students and researchers in other sciences.

We are grateful to the readers for their comments on this book over the past 40 years and would welcome any questions, comments, and suggestions. You can communicate with Vijay K. Rohatgi at [email protected] and with A. K. Md. Ehsanes Saleh at [email protected]

Vijay K. Rohatgi

A. K. Md. Ehsanes Saleh

Solana Beach, CAOttawa, Canada


There is a lot that is different about this second edition. First, there is a co-author without whose help this revision would not have been possible. Second, we have benefited from countless letters from readers and colleagues who have pointed out errors and omissions and have made valuable suggestions over the past 25 years. These communications make this revision worth the effort. Third, we have tried to update the content of the book while striving to preserve the character and spirit of the first edition.

Here are some of the numerous changes that have been made.

The Introduction section has been removed. We have also removed Chapter 14 on sequential statistical inference.

Many parts of the book have gone substantial rewriting. For example,

Chapter 4

has many changes, such as inclusion of exchangeability. In

Chapter 3

, an introduction to characteristic functions has been added. In

Chapter 5

some new distributions have been added while in

Chapter 6

there have been many changes in proofs.

The statistical inference part of the book (

Chapters 8



) has been updated. Thus in

Chapter 8

we have expanded the coverage of invariance and have included discussions of ancillary statistics and conjugate prior distributions.

Similar changes have been made in

Chapter 9

. A new section on locally most powerful tests has been added.

Chapter 11

has been greatly revised and a discussion of invariant confidence intervals has been added.

Chapter 13

has been completely rewritten in the light of increased emphasis on nonparametric inference. We have expanded the discussion of {/-statistics. Later sections show the connection between commonly used tests and {-statistics.


Chapter 12

, the notation has been changed to confirm to the current convention.

Many problems and examples have been added.

More figures have been added to illustrate examples and proofs.

Answers to selected problems have been provided.

We are truly grateful to the readers of the first edition for countless comments and suggestions and hope we will continue to hear from them about this edition.

Special thanks are due Ms. Gillian Murray for her superb word processing of the manuscript, and Dr. Indar Bhatia for figures that appear in the text. Dr. Bhatia spent countless hours preparing the diagrams for publication. We also acknowledge the assistance of Dr. K. Selvavel.

Vijay K. Rohatgi

A. K. Md. Ehsanes Saleh


This book on probability theory and mathematical statistics is designed for a three-quarter course meeting 4 hours per week or a two-semester course meeting 3 hours per week. It is designed primarily for advanced seniors and beginning graduate students in mathematics, but it can also be used by students in physics and engineering with strong mathematical backgrounds. Let me emphasize that this is a mathematics text and not a “cookbook.” It should not be used as a text for service courses.

The mathematics prerequisites for this book are modest. It is assumed that the reader has had basic courses in set theory and linear algebra and a solid course in advanced calculus. No prior knowledge of probability and/or statistics is assumed.

My aim is to provide a solid and well-balanced introduction to probability theory and mathematical statistics. It is assumed that students who wish to do graduate work in probability theory and mathematical statistics will be taking, concurrently with this course, a measure-theoretic course in analysis if they have not already had one. These students can go on to take advanced-level courses in probability theory or mathematical statistics after completing this course.

This book consists of essentially three parts, although no such formal divisions are designated in the text. The first part consists of Chapters 1 through 6, which form the core of the probability portion of the course. The second part, Chapters 7 through 11, covers the foundations of statistical inference. The third part consists of the remaining three chapters on special topics. For course sequences that separate probability and mathematical statistics, the first part of the book can be used for a course in probability theory, followed by a course in mathematical statistics based on the second part and, possibly, one or more chapters on special topics.

The reader will find here a wealth of material. Although the topics covered are fairly conventional, the discussions and special topics included are not. Many presentations give far more depth than is usually the case in a book at this level. Some special features of the book are the following:

A well-referenced chapter on the preliminaries.

About 550 problems, over 350 worked-out examples, about 200 remarks, and about 150 references.

An advance warning to reader wherever the details become too involved. They can skip the later portion of the section in question on first reading without destroying the continuity in any way.

Many results on characterizations of distributions (

Chapter 5


Proof of the central limit theorem by the method of operators and proof of the strong law of large numbers (

Chapter 6


A section on minimal sufficient statistics (

Chapter 8


A chapter on special tests (

Chapter 10


A careful presentation of the theory of confidence intervals, including Bayesian intervals and shortest-length confidence intervals (

Chapter 11


A chapter on the general linear hypothesis, which carries linear models through to their use in basic analysis of variance (

Chapter 12


Sections on nonparametric estimation and robustness (

Chapter 13


Two sections on sequential estimation (Chapter 14).

The contents of this book were used in a 1 -year (two-semester) course that I taught three times at the Catholic University of America and once in a three-quarter course at Bowling Green State University. In the fall of 1973 my colleague, Professor Eugene Lukacs, taught the first quarter of this same course on the basis of my notes, which eventually became this book. I have always been able to cover this book (with few omissions) in a 1-year course, lecturing 3 hours a week. An hour-long problem session every week is conducted by a senior graduate student.

In a book of this size there are bound to be some misprints, errors, and ambiguities of presentation. I shall be grateful to any reader who brings these to my attention.

V. K. Rohatgi

Bowling Green, OhioFebruary 1975


We take this opportunity to thank many correspondents whose comments and criticisms led to improvements in the Third Edition. The list below is far from complete since it does not include the names of countless students whose reactions to the book as a text helped the authors in this revised edition. We apologize to those whose names may have been inadvertently omitted from the list because we were not diligent enough to keep a complete record of all the correspondence. For the third edition we wish to thank Professors Yue-Cune Chang, Anirban Das Gupta, A. G. Pathak, Arno Weiershauser, and many other readers who sent their questions and comments. We also wish to acknowledge the assistance of Dr. Pooplasingam Sivakumar in preparation of the manuscript. For the second edition: Barry Arnold, Lennart Bondesson, Harry Cohn, Frank Connonito, Emad El-Neweihi, Ulrich Faigle, Pier Alda Ferrari, Martin Feuerrnan, Xavier Fernando, Z. Govindarajulu, Arjun Gupta, Hassein Hamedani, Thomas Hem, Jin-Sheng Huang, Bill Hudson, Barthel Huff, V. S. Huzurbazar, B. K. Kale, Sam Kotz, Bansi Lal, Sri Gopal Mohanty, M. V. Moorthy, True Nguyen, Tom O'Connor, A. G. Pathak, Edsel Pena, S. Perng, Madan Puri, Prem Puri, J. S. Rao, Bill Raser, Andrew Rukhin, K. Selvavel, Rajinder Singh, R. J. Tomkins; for the first edition, Ralph Baty, Ralph Bradley, Eugene Lukacs, Kae Lea Main, Tom and Carol O'Connor, M. S. Scott Jr., J. Sethuraman, Beatrice Shube, Jeff Spielman, and Robert Tortora.

We thank the publishers of the American Mathematical Monthly, the SIAM Review, and the American Statistician for permission to include many examples and problems that appeared in these journals. Thanks are also due to the following for permission to include tables: Professors E. S. Pearson and L. R. Verdooren (Table ST11), Harvard University Press (Table ST1), Hafner Press (Table ST3), Iowa State University Press (Table ST5), Rand Corporation (Table ST6), the American Statistical Association (Tables ST7 and ST10), the Institute of Mathematical Statistics (Tables ST8 and ST9), Charles Griffin & Co., Ltd. (Tables ST12 and ST13), and John Wiley & Sons (Tables ST1, ST2, ST4, ST10, and ST11).


This book is divided into 13 chapters, numbered 1 through 13. Each chapter is divided into several sections. Lemmas, theorems, equations, definitions, remarks, figures, and so on, are numbered consecutively within each section. Thus Theorem i.j.k refers to the kth theorem in Section j of Chapter i, Section i.j refers to the jth section of Chapter i, and so on. Theorem j refers to the jth theorem of the section in which it appears. A similar convention is used for equations except that equation numbers are enclosed in parentheses. Each section is followed by a set of problems for which the same numbering system is used.

References are given at the end of the book and are denoted in the text by numbers enclosed in square brackets, [ ]. If a citation is to a book, the notation ([i, p. j]) refers to the jth page of the reference numbered [i].

A word about the proofs of results stated without proof in this book. If a reference appears immediately following or preceding the statement of a result, it generally means that the proof is beyond the scope of this text. If no reference is given, it indicates that the proof is left to the reader. Sometimes the reader is asked to supply the proof as a problem.