Probability via Expectation

369 Pages·2000·22.944 MB·English

by  Peter Whittle (auth.)

Springer Texts in Statistics
Advisors:
George Casella Stephen Fienberg Ingram Ollan
Springer Science+ Business Media, LLC
Springer Texts in Statistics
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Bilodeau and Brenner: Theory of Multivariate Statistics
Blom: Probability and Statistics: Theory and Applications
Brockwell and Davis: An Introduction to Times Series and Forecasting
Chow and Teicher: Probability Theory: Independence, Interchangeability,
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Christensen: Plane Answers to Complex Questions: The Theory of Linear
Models, Second Edition
Christensen: Linear Models for Multivariate, Time Series, and Spatial Data
Christensen: Log-Linear Models and Logistic Regression, Second Edition
Creighton: A First Course in Probability Models and Statistical Inference
Dean and Voss: Design and Analysis of Experiments
du Toit, Steyn, and Stumpf Graphical Exploratory Data Analysis
Durrett: Essentials of Stochastic Processes
Edwards: Introduction to Graphical Modelling
Finkelstein and Levin: Statistics for Lawyers
Flury: A First Course in Multivariate Statistics
Jobson: Applied Multivariate Data Analysis, Volume I: Regression and
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Jobson: Applied Multivariate Data Analysis, Volume II: Categorical and
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Kalbfleisch: Probability and Statistical Inference, Volume I: Probability,
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Kalbfleisch: Probability and Statistical Inference, Volume II: Statistical
Inference, Second Edition
Karr: Probability
Keyjitz: Applied Mathematical Demography, Second Edition
Kiefer: Introduction to Statistical Inference
Kokoska and Nevison: Statistical Tables and Formulae
Kulkarni: Modeling, Analysis, Design, and Control of Stochastic Systems
Lehmann: Elements of Large-Sample Theory
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Lehmann and Casella: Theory of Point Estimation, Second Edition
Lindman: Analysis of Variance in Experimental Design
Lindsey: Applying Generalized Linear Models
Madansky: Prescriptions for Working Statisticians
McPherson: Statistics in Scientific Investigation: Its Basis, Application, and
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Mueller: Basic Principles of Structural Equation Modeling: An Introduction to
LISREL and EQS
(continued after index)
Peter Whittle
Probability via
Expectation
Fourth Edition
With 22 lllustrations
, Springer
Peter Whittle
Department of Pure Mathematics
and Mathematical Statistics
University of Cambridge
Cambridge CB2 lSB
Editorial Board
George Casella Stephen Fienberg Ingram Olkin
Department of Biometrics Department of Statistics Department of Statistics
Cornell University Carnegie Mellon University Stanford University
Ithaca, NY 14853-78 Pittsburgh, PA 15213-3890 Stanford, CA 94305
USA USA USA
Library of Congress Cata10ging-in-Publication Data
Whittle, Peter.
Probability via expectation / Peter Whittle. - 4th ed.
p. cm. - (Springer texts in statistics)
Inc1udes bibliographical references and index.
ISBN 978-1-4612-6795-9 ISBN 978-1-4612-0509-8 (eBook)
DOI 10.1007/978-1-4612-0509-8
1. Probabilities. I. Title. II. Series.
QA273.W59 2000
519.2-dc21 99-053569
Printed on acid-free paper.
Russian trans1ation, Nauka, 1982. Second edition, Wiley, 1976. First edition, Penguin, 1970.
© 2000 Springer Science+Business Media New York
Originally published by Springer-VeriagNew York in 2000
Softcover reprint of the hardcover 4th edition 2000
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ISBN 978-1-4612-6795-9
To my parents
Preface to the Fourth Edition
The third edition of 1992 constituted a major reworking of the original text, and
the preface to that edition still represents my position on the issues that stimulated
me first to write. The present edition contains a number of minor modifications
and corrections, but its principal innovation is the addition of material on dynamic
programming, optimal allocation, option pricing and large deviations. These are
substantial topics, but ones into which one can gain an insight with less labour
than is generally thought. They all involve the expectation concept in an essential
fashion, even the treatment of option pricing, which seems initially to forswear
expectation in favour of an arbitrage criterion.
I am grateful to readers and to Springer-Verlag for their continuing interest in
the approach taken in this work.
Peter Whittle
Preface to the Third Edition
This book is a complete revision of the earlier work Probability which appeared
in 1970. While revised so radically and incorporating so much new material as to
amount to a new text, it preserves both the aim and the approach of the original.
That aim was stated as the provision of a 'first text in probability, demanding
a reasonable but not extensive knowledge of mathematics, and taking the reader
to what one might describe as a good intermediate level' . In doing so it attempted
to break away from stereotyped applications, and consider applications of a more
novel and significant character.
The particular novelty of the approach was that expectation was taken as the
prime concept, and the concept of expectation axiomatized rather than that of
a probability measure. In the preface to the original text of 1970 (reproduced
below, together with that to the Russian edition of 1982) I listed what I saw as
the advantages of the approach in as unlaboured a fashion as I could. I also took
the view that the text rather than the author should persuade, and left the text
to speak for itself. It has, indeed, stimulated a steady interest, to the point that
Springer-Verlag has now commissioned this complete reworking.
In re-examining the approach after this lapse of time I find it more persuasive
than ever. Indeed, I believe that the natural flow of the argument is now more
evident to me, and that this revised version is much more successful in tracing
that flow from initial premises to surprisingly advanced conclusions. At the risk I
fear most-of labouring the argument-I would briefly list the advantages of the
expectation approach as follows.
(i) It permits a more economic and natural treatment at the elementary level.
(ii) It opens an immediate door to applications, because the quantity of interest
in many applications is just an expectation.
x Preface to the Third Edition
(iii) Precisely for this last reason, one can discuss applications of genuine interest
with very little preliminary development of theory. On the other hand, one
also finds that a natural unrolling of ideas leads to the development of theory
almost of itself.
(iv) The approach is an intuitive one, in that people have a well-developed in
tuition for the concept of an average. Of course, what is found 'intuitive'
depends on one's experience, but people with a background in the physical
sciences have certainly taken readily to the approach. Historically, the early
analysts of games of chance found the question 'What is a fair price for
entering a garneT quite as natural as 'What is the probability of winning itT
We make some historical observations in Section 3.4.
(v) The treatment is the natural one at an advanced level. However, as noted in
the preface to Probability, here we do not need to make a case-the accepted
concepts and techniques of weak convergence and of generalized processes
are characterized wholly in terms of expectation.
(vi) Much conventional presentation of probability theory is distorted by a preoc
cupation with measure-theoretic concepts which is in a sense premature and
irrelevant. These concepts (or some equivalent of them) cannot be avoided
indefinitely. However, in the expectation approach, they find their place at
the natural stage.
(vii) On the other hand, a concept which is notably and remarkably absent from
conventional treatments is that of convexity. (Remarkable, because convex
ity is a probabilistic concept, and, in optimization theory, the necessary
invocations of convexity and of probabilistic ideas are intimately related.) In
the expectation approach convexity indeed emerges as an inevitable central
concept.
(viii) Finally, in the expectation approach, classical probability and the probability
of quantum theory are seen to differ only in a modification of the axioms - a
modification rich in consequences, but succinctly expressible.
The reader can be reassured that the book covers at least the material that
would be found in any modern text of this level, and will leave him at least as
well equipped in conventional senses as these. The difference is one of order and
emphasis, although this cannot be dismissed, since it gives the book its point.
The enhanced role of convexity has already been mentioned. The concept of least
square approximation, fundamental in so many nonprobabilistic contexts, is found
to pervade the treatment. In the discussion of stochastic processes one is led to give
much greater importance than usual to the backward equation, which reveals both
the generator of the process and another all-prevading concept, that of a martingale.
The conventions on the numbering of equations, etc. are not quite uniform,
but are the most economical. Sections and equations are numbered consecutively
through the chapter, so that a reference to 'Section 2' means Section 2 of the
current chapter, whereas a reference to 'Section 4.2' is to Section 2 of Chapter 4.
Correspondingly for equations. Figures are also numbered consecutively through a
chapter, but always carry a chapter label; e.g. 'Fig. 12.3'. Theorems are numbered
Preface to the Third Edition xi
consecutively through a section, and always carry full chapter/section/number
label; e.g. 'Theorem 5.3.2' for Theorem 2 of Section 5.3. Exercises are numbered
consecutively through a section, and are given a chapter/section reference (e.g.
Exercise 10.9.2) only when referred to from another section.
I am grateful to David Stirzaker for some very apt historical observations and
references, also to Roland Tegeder for helpful discussion of the final two sections.
The work was supported in various phases by the Esso Petroleum Company Ltd.
and by the United Kingdom Science and Engineering Research Council. I am most
grateful to these two bodies.
Peter Whittle
Preface to the Russian Edition of
Probability (1982)
When this text was published in 1970 I was aware of its unorthodoxy, and uncertain
of its reception. Nevertheless, I was resolved to let it speak for itself, and not to
advocate further the case there presented. This was partly because of an intrinsic
unwillingness to propagandize, and partly because of a conviction that an approach
which I (in company with Huygens and other early authors) found so natural would
ultimately need no advocate. It has then been a great pleasure to me that others
have also shared this latter view and have written in complimentary terms to say
so. However, the decision of the 'Nauka' Publishing House to prepare a Russian
edition implies the compliment I value most, in view of the quite special role
Russian authors have played in the development of the theory of probability.
I have taken the opportunity to correct some minor errors kindly pointed out to
me by readers, but the work is otherwise unrevised. My sincere thanks are due to
Professor N. Gamkrelidze for bringing to the unrewarding task of translation, not
only high professional competence, but even enthusiasm.
Peter Whittle