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Elements of Probability and Statistics: An Introduction to Probability with de Finetti's Approach and to Bayesian Statistics PDF

246 Pages·2016·3.67 MB·English
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Francesca Biagini Massimo Campanino Elements of Probability T T and Statistics X X An Introduction to Probability E E with de Finetti’s Approach and to Bayesian Statistics TT II NN UU UNITEXT - La Matematica per il 3+2 Volume 98 Editor-in-chief A. Quarteroni Series editors L. Ambrosio P. Biscari C. Ciliberto M. Ledoux W.J. Runggaldier More information about this series at http://www.springer.com/series/5418 Francesca Biagini Massimo Campanino (cid:129) Elements of Probability and Statistics An Introduction to Probability ’ with de Finetti s Approach and to Bayesian Statistics 123 Francesca Biagini Massimo Campanino Department ofMathematics Department ofMathematics Ludwig-Maximilians-Universität Universitàdi Bologna Munich Bologna Germany Italy ISSN 2038-5722 ISSN 2038-5757 (electronic) UNITEXT- La Matematica peril3+2 ISBN978-3-319-07253-1 ISBN978-3-319-07254-8 (eBook) DOI 10.1007/978-3-319-07254-8 LibraryofCongressControlNumber:2015958841 TranslationfromtheItalianlanguageedition:ElementidiProbabilitàeStatisticadiFrancescaBiaginie MassimoCampanino,©Springer-VerlagItalia,Milano2006.Allrightsreserved. ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Coverdesign:SimonaColombo,GiochidiGrafica,Milano,Italy Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Nous ne possédons une ligne, un surface, un volume que si notre amour l’occupe. M. Proust To Thilo and Oskar Francesca Biagini To my brother Vittorio Massimo Campanino Preface This book is based on the lectures notes for the course, Probability and Mathematical Statistics, taught for many years by one of the authors (M.C.) and then,dividedintotwosections,bybothauthorsattheUniversityofBologna(Italy). We follow the approach of de Finetti, see de Finetti [1] for a complete detailed exposition. Although de Finetti [1] was conceived as a textbook of probability for mathematicsstudents,itwasalsomeanttoillustratethepointofviewoftheauthor on the foundations of probability and mathematical statistics and discuss it in relation to prevalent approaches, resulting often of difficult access for beginners. This was the main reason that prompted us to arrange the lectures notes of our courses into a more organic way and to write a textbook for an initial class on probability and mathematical statistics. The first five chapters are devoted to elementary probability. After that in the next three chapters we develop some elements of Markov chains in discrete and continuous time also in connection with queueing processes, and introduce basic conceptsin mathematical statistics in theBayesian approach. Then we proposesix chapters of exercises, which cover most of the topics treated in the theoretical part. In the appendices we have inserted summary schemes and complementary topics(twoproofsofStirlingformula).Wealsoinformallyrecallsomeelementsof calculus, as this has often proved useful for the students. This book offers a comprehensive but concise introduction to probability and mathematicalstatisticswithoutrequiringnotionsofmeasuretheory;henceitcanbe used in basic classes on probability for mathematics students and is particularly suitable for computer science, physics and engineering students. ix x Preface WearegratefultoSpringerforallowingustopublishtheEnglishversionofthe book. We wish to thank Elisa Canova, Alessandra Cretarola, Nicola Mezzetti and Quirin Vogel for their fundamental help with latex, for both the Italian and the English version. Munich Francesca Biagini Bologna Massimo Campanino June 2015 Contents Part I Probability 1 Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Expectation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Probability of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Uniform Distribution on Partitions . . . . . . . . . . . . . . . . . . . . . 12 1.6 Conditional Probability and Expectation . . . . . . . . . . . . . . . . . 14 1.7 Formula of Composite Expectation and Probability. . . . . . . . . . 15 1.8 Formula of Total Expectation and Total Probability . . . . . . . . . 16 1.9 Bayes Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.10 Correlation Between Events. . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.11 Stochastic Independence and Constituents. . . . . . . . . . . . . . . . 20 1.12 Covariance and Variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.13 Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.14 Chebychev’s Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.15 Weak Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Discrete Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1 Random Numbers with Discrete Distribution. . . . . . . . . . . . . . 27 2.2 Bernoulli Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Binomial Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 Poisson Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.7 Independence of Partitions. . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.8 Generalized Bernoulli Scheme. . . . . . . . . . . . . . . . . . . . . . . . 33 2.9 Multinomial Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.10 Stochastic Independence for Random Numbers with Discrete distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.11 Joint Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 xi

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