, AN INTRODUCTION TO
MATHEMATICAL STATISTICS
AND I TS A PPLICATIONS
Fifth Edition
Richard J. Larsen
Vanderbilt University
Morris L. Marx
University of West Florida
Prentice Hall
Boston Columbus Indianapolis New York San Francisco
Upper Saddle River Amsterdam Cape Town Dubai
London Madrid Milan Munich Paris Montréal
Toronto Delhi Mexico City São Paulo Sydney
Hong Kong Seoul Singapore Taipei Tokyo
,Editor in Chief: Deirdre Lynch
Acquisitions Editor: Christopher Cummings
Associate Editor: Christina Lepre
Assistant Editor: Dana Jones
Senior Managing Editor: Karen Wernholm
Associate Managing Editor: Tamela Ambush
Senior Production Project Manager: Peggy McMahon
Senior Design Supervisor: Andrea Nix
Cover Design: Beth Paquin
Interior Design: Tamara Newnam
Marketing Manager: Alex Gay
Marketing Assistant: Kathleen DeChavez
Senior Author Support/Technology Specialist: Joe Vetere
Manufacturing Manager: Evelyn Beaton
Senior Manufacturing Buyer: Carol Melville
Production Coordination, Technical Illustrations, and Composition: Integra Software Services, Inc.
Cover Photo: © Jason Reed/Getty Images
Many of the designations used by manufacturers and sellers to distinguish their products are claimed as
trademarks. Where those designations appear in this book, and Pearson was aware of a trademark
claim, the designations have been printed in initial caps or all caps.
Library of Congress Cataloging-in-Publication Data
Larsen, Richard J.
An introduction to mathematical statistics and its applications /
Richard J. Larsen, Morris L. Marx.—5th ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-321-69394-5
1. Mathematical statistics—Textbooks. I. Marx, Morris L. II. Title.
QA276.L314 2012
519.5—dc22
2010001387
Copyright © 2012, 2006, 2001, 1986, and 1981 by Pearson Education, Inc. All rights reserved. No part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any
means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America. For information on obtaining
permission for use of material in this work, please submit a written request to Pearson Education, Inc.,
Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request
to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm.
1 2 3 4 5 6 7 8 9 10—EB—14 13 12 11 10
ISBN-13: 978-0-321-69394-5
ISBN-10: 0-321-69394-9
, Table of Contents
Preface viii
1 Introduction 1
1.1 An Overview 1
1.2 Some Examples 2
1.3 A Brief History 7
1.4 A Chapter Summary 14
2 Probability 16
2.1 Introduction 16
2.2 Sample Spaces and the Algebra of Sets 18
2.3 The Probability Function 27
2.4 Conditional Probability 32
2.5 Independence 53
2.6 Combinatorics 67
2.7 Combinatorial Probability 90
2.8 Taking a Second Look at Statistics (Monte Carlo Techniques) 99
3 Random Variables 102
3.1 Introduction 102
3.2 Binomial and Hypergeometric Probabilities 103
3.3 Discrete Random Variables 118
3.4 Continuous Random Variables 129
3.5 Expected Values 139
3.6 The Variance 155
3.7 Joint Densities 162
3.8 Transforming and Combining Random Variables 176
3.9 Further Properties of the Mean and Variance 183
3.10 Order Statistics 193
3.11 Conditional Densities 200
3.12 Moment-Generating Functions 207
3.13 Taking a Second Look at Statistics (Interpreting Means) 216
Appendix 3.A.1 Minitab Applications 218
iii
MATHEMATICAL STATISTICS
AND I TS A PPLICATIONS
Fifth Edition
Richard J. Larsen
Vanderbilt University
Morris L. Marx
University of West Florida
Prentice Hall
Boston Columbus Indianapolis New York San Francisco
Upper Saddle River Amsterdam Cape Town Dubai
London Madrid Milan Munich Paris Montréal
Toronto Delhi Mexico City São Paulo Sydney
Hong Kong Seoul Singapore Taipei Tokyo
,Editor in Chief: Deirdre Lynch
Acquisitions Editor: Christopher Cummings
Associate Editor: Christina Lepre
Assistant Editor: Dana Jones
Senior Managing Editor: Karen Wernholm
Associate Managing Editor: Tamela Ambush
Senior Production Project Manager: Peggy McMahon
Senior Design Supervisor: Andrea Nix
Cover Design: Beth Paquin
Interior Design: Tamara Newnam
Marketing Manager: Alex Gay
Marketing Assistant: Kathleen DeChavez
Senior Author Support/Technology Specialist: Joe Vetere
Manufacturing Manager: Evelyn Beaton
Senior Manufacturing Buyer: Carol Melville
Production Coordination, Technical Illustrations, and Composition: Integra Software Services, Inc.
Cover Photo: © Jason Reed/Getty Images
Many of the designations used by manufacturers and sellers to distinguish their products are claimed as
trademarks. Where those designations appear in this book, and Pearson was aware of a trademark
claim, the designations have been printed in initial caps or all caps.
Library of Congress Cataloging-in-Publication Data
Larsen, Richard J.
An introduction to mathematical statistics and its applications /
Richard J. Larsen, Morris L. Marx.—5th ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-321-69394-5
1. Mathematical statistics—Textbooks. I. Marx, Morris L. II. Title.
QA276.L314 2012
519.5—dc22
2010001387
Copyright © 2012, 2006, 2001, 1986, and 1981 by Pearson Education, Inc. All rights reserved. No part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any
means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America. For information on obtaining
permission for use of material in this work, please submit a written request to Pearson Education, Inc.,
Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request
to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm.
1 2 3 4 5 6 7 8 9 10—EB—14 13 12 11 10
ISBN-13: 978-0-321-69394-5
ISBN-10: 0-321-69394-9
, Table of Contents
Preface viii
1 Introduction 1
1.1 An Overview 1
1.2 Some Examples 2
1.3 A Brief History 7
1.4 A Chapter Summary 14
2 Probability 16
2.1 Introduction 16
2.2 Sample Spaces and the Algebra of Sets 18
2.3 The Probability Function 27
2.4 Conditional Probability 32
2.5 Independence 53
2.6 Combinatorics 67
2.7 Combinatorial Probability 90
2.8 Taking a Second Look at Statistics (Monte Carlo Techniques) 99
3 Random Variables 102
3.1 Introduction 102
3.2 Binomial and Hypergeometric Probabilities 103
3.3 Discrete Random Variables 118
3.4 Continuous Random Variables 129
3.5 Expected Values 139
3.6 The Variance 155
3.7 Joint Densities 162
3.8 Transforming and Combining Random Variables 176
3.9 Further Properties of the Mean and Variance 183
3.10 Order Statistics 193
3.11 Conditional Densities 200
3.12 Moment-Generating Functions 207
3.13 Taking a Second Look at Statistics (Interpreting Means) 216
Appendix 3.A.1 Minitab Applications 218
iii
