Probability and Statistical Models Foundations for Problems in Relia
With an emphasis on models and techniques, this textbook introduces many of the fundamental concepts of stochastic modeling that are now a vital component of almost every scientific investigation. These models form the basis of well-known parametric lifet
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Wei-Bin Zeng
Yanhong Wu
Probability and Statistical Models Foundations for Problems in Reliability and Financial Mathematics
Birkh¨auser
Arjun K. Gupta Department of Mathematics and Statistics Bowling Green State University Bowling Green, OH 43403 USA [email protected]
Yanhong Wu Department of Mathematics California State University Stanislaus One University Circle Turlock, CA 95382 USA [email protected]
Wei-Bin Zeng Department of Mathematics University of Louisville 328 Natural Sciences Building Louisville, KY 40292 USA [email protected]
ISBN 978-0-8176-4986-9 e-ISBN 978-0-8176-4987-6 DOI 10.1007/978-0-8176-4987-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010934423 Mathematics Subject Classification (2010): Primary: 60-01, 62-01, 91-01; Secondary: 60K05, 60K10, 90B25, 91B30, 91B24, 91G40, 91G20 c Springer ScienceCBusiness Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer ScienceCBusiness Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper www.birkhauser-science.com
For Arhaan, Danwen, and Lin
Preface
Probability models are now a vital component of every scientific investigation. This book is intended to introduce basic ideas in stochastic modeling, with emphasis on models and techniques. These models lead to well-known parametric lifetime distributions, such as exponential, Weibull, and gamma distributions, as well as the change-point and mixture models. They also motivate us to consider more general notions of nonparametric lifetime distribution classes. Particular attention has been paid to their applications in reliability, insurance mathematics, and economics. The following topics are the focus in this volume: 1. 2. 3. 4. 5. 6. 7.
Exponential Distributions and the Poisson Process; Parametric Lifetime Distributions; Nonparametric Lifetime Distribution Classes; Multivariate Exponential Extensions; Association and Dependence; Renewal Theory; Applications to Reliability, Insurance, Finance, and Credit Risk.
Chapter 1 provides notation and basic results in probability theory that are needed in the consequent chapters. Chapters 2 and 3 are devoted to models related to exponential distribution and Poisson processes. Particular attentions is paid to the characterizations of exponential distribution and the Poisson process. Two of the most important properties that characterize exponential distribution: the lack of
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