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

246 Pages·2016·3.683 MB·English

by  Francesca Biagini, Massimo Campanino (auth.)

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