Introduction to probability

548 Pages·2014·4.376 MB·English

by  George G Roussas

Introduction to
Probability
Second Edition
This page is intentionally left blank
Introduction to
Probability
Second Edition
George G. Roussas
Department of Statistics
University of California, Davis
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK
OXFORD • PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE
SYDNEY • TOKYO
Academic Press is an imprint of Elsevier
Academic Press is an imprint of Elsevier
The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK
Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands
225 Wyman Street, Waltham, MA 02451, USA
525 B Street, Suite 1800, San Diego, CA 92101-4495, USA
Second edition 2014
Copyright © 2014, 2007 Elsevier 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 Permissions may be sought directly from
Elsevier’s Science & Technology Rights Department in Oxford, UK: phone ( 44) (0) 1865
+
843830; fax ( 44) (0) 1865 853333; email: [email protected]. Alternatively you can
+
submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/per-
missions, and selecting Obtaining permission to use Elsevier material.
Notice
No responsibility is assumed by the publisher for any injury and/or damage to persons or prop-
erty as a matter of products liability, negligence or otherwise, or from any use or operation of
any methods, products, instructions or ideas contained in the material herein. Because of rapid
advances in the medical sciences, in particular, independent verification of diagnoses and drug
dosages should be made.
Library of Congress Cataloging-in-Publication Data
Roussas, George G., author.
Introduction to probability / by George G. Roussas. -- Second edition.
pages cm
Includes bibliographical references and index.
ISBN 978-0-12-800041-0
1. Probabilities--Textbooks. I. Title.
QA273.R8647 2014
519.2--dc23
2013036799
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
For information on all Academic Press publications
visit our web site at store.elsevier.com
Printed and bound in USA
14 15 16 17 18 10 9 8 7 6 5 4 3 2 1
ISBN: 978-0-12-800041-0
To the memory of David Blackwell and Lucien Le Cam
This page is intentionally left blank
Contents
Preface ��������������������������������������������������������������������������������������������������������xi
CHAPTER 1� Some Motivating Examples ����������������������������������������������1
CHAPTER 2� Some Fundamental Concepts �������������������������������������������7
2�1 Some Fundamental Concepts .........................................................7
2�2 Some Fundamental Results ...........................................................11
2�3 Random Variables .........................................................................19
2�4 Basic Concepts and Results in Counting ......................................23
CHAPTER 3� The Concept of Probability and Basic Results ������������31
3�1 Definition of Probability ...............................................................31
3�2 Some Basic Properties and Results ...............................................35
3�3 Distribution of a Random Variable ...............................................44
CHAPTER 4� Conditional Probability and Independence ������������������55
4�1 Conditional Probability and Related Results ................................55
4�2 Independent Events and Related Results ......................................66
CHAPTER 5� Numerical Characteristics of a Random Variable �������79
5�1 Expectation, Variance, and Moment-Generating Function of a
Random Variable ...........................................................................79
5�2 Some Probability Inequalities .......................................................89
5�3 Median and Mode of a Random Variable .....................................91
CHAPTER 6� Some Special Distributions ��������������������������������������������99
6�1 Some Special Discrete Distributions ............................................99
6�1�1 Binomial Distribution ........................................................99
6�1�2 Geometric Distribution ....................................................104
6�1�3 Poisson Distribution ........................................................106
6�1�4 Hypergeometric Distribution ...........................................108
6�2 Some Special Continuous Distributions .....................................116
6�2�1 Gamma Distribution ........................................................117
6�2�2 Negative Exponential Distribution ..................................119
6�2�3 Chi-Square Distribution ...................................................121
6�2�4 Normal Distribution .........................................................121
6�2�5 Uniform (or Rectangular) Distribution ............................126
6�2�6 The basics of the Central Limit Theorem(CLT) ..............134
vii
viii Contents
CHAPTER 7� Joint Probability Density Function of Two Random
Variables and Related Quantities ��������������������������������137
7�1 Joint d.f. and Joint p.d.f. of Two Random Variables ...................137
7�2 Marginal and Conditional p.d.f.'s, Conditional Expectation
and Variance ................................................................................149
CHAPTER 8� Joint Moment-Generating Function, Covariance,
and Correlation Coefficient of Two Random
Variables �������������������������������������������������������������������������163
8�1 The Joint m.g.f. of Two Random Variables ................................163
8�2 Covariance and Correlation Coefficient of Two
Random Variables .......................................................................167
8�3 Proof of Theorem 1, Some Further Results ................................174
CHAPTER 9� Some Generalizations to k Random Variables,
and Three Multivariate Distributions ��������������������������179
9�1 Joint Distribution of k Random Variables
and Related Quantities ................................................................179
9�2 Multinomial Distribution ............................................................183
9�3 Bivariate Normal Distribution ....................................................189
9�4 Multivariate Normal Distribution ...............................................198
CHAPTER 10� I ndependence of Random Variables and Some
Applications ������������������������������������������������������������������201
10�1 Independence of Random Variables and Criteria of
Independence ..............................................................................201
10�2 The Reproductive Property of Certain Distributions ..................213
10�3 Distribution of the Sample Variance under Normality................222
CHAPTER 11� T ransformation of Random Variables������������������������225
11�1 Transforming a Single Random Variable ....................................225
11�2 Transforming Two or More Random Variables ..........................231
11�3 Linear Transformations ...............................................................245
11�4 The Probability Integral Transform ............................................253
11�5 Order Statistics ............................................................................255
CHAPTER 12� T wo Modes of Convergence, the Weak Law
of Large Numbers, the Central Limit Theorem,
and Further Results �����������������������������������������������������265
12�1 Convergence in Distribution and in Probability .........................265
12�2 The Weak Law of Large Numbers and the Central
Limit Theorem ............................................................................271