Probability: an Introduction

Probability An Introduction Probability An Introduction Second Edition Geoffrey Grimmett University of Cambridge Dominic Welsh University of Oxford 3 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Geoffrey Grimmett & Dominic Welsh 2014 The moral rights of the authors have been asserted First Edition published in 1986 Second Edition published in 2014 Impression: 1 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, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2014938231 ISBN 978–0–19–870996–1 (hbk.) ISBN 978–0–19–870997–8 (pbk.) Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Preface to the second edition Probability theory is now fully established as a crossroads discipline in mathematical science. Its connections across pure and applied mathematics are in excellent health. In its role as the ‘science of risk’, it is a lynchpin of political, economic, and social science. Never before have so many students in schools and universities been exposed to the basics of probability. This introductory text on probability is designed for first and second year mathematics students. It is based upon courses given at the Universities of Bristol, Cambridge, and Oxford. Broadly speaking, we cover the usual material, but we hope that our account will have certain special attractions for the reader and we shall say what these may be in a moment. The first eight chapters form a course in basic probability, being an account of events, random variables, and distributions—we treat discrete and continuous random variables separately— together with simple versions of the law of large numbers and the central limit theorem. There is an account of moment generating functions and their applications. The next three chapters are about branching processes, random walks, and continuous-time random processes such as the Poisson process. We hope that these chapters are adequate at this level and are suitable appetizers for further courses in applied probability and random processes. The final chapter is devoted to Markov chains in discrete time. As in the first edition, this text is divided into three sections: (A) Probability, (B) Further Probability, and (C) Random Processes. We hope thus to indicate two things. First, the probability in Part A seems to us to be core material for first-year students, whereas the material in Part B is somewhat more difficult. Secondly, although random processes are collected together in the final four chapters, they may well be introduced much earlier in the course. The chapters on branching processes and random walks might be studied after Chapter 5, and the chapter on continuous-time processes after Chapter 6. The chapter on Markov chains can in addition be used as the basis for a free-standing 12–16 lecture course. The major difference of substance between the first and second editions of this text is the new Chapter 12 on Markov chains. This chapter is a self-contained account of discrete-time chains, culminating in a proper account of the convergencetheorem. Numerous lesser changes and additions have been made to the text in response to the evolution of course syllabuses and of our perceptionsof the needs of readers. These include more explicit accounts of geometrical probability, indicator functions, the Markov and Jensen inequalities, the multivariate normal distribution, and Cramér’s large deviation theorem, together with further exercises and problems often taken from recent examination papers at our home universities. We have two major aims: to be concise, and to be honest about mathematical rigour. Some will say that this book reads like a set of lecture notes. We would not regard this as entirely unfair; indeed a principal reason for writing it was that we believe that most students benefit more from possessing a compact accountof the subject in 250 printed pages or so (at a suitable price) than a diffuse account of 400 or more pages. Most undergraduates learn probability vi Preface to the second edition theory by attending lectures, at which they may take copious and occasionally incorrect notes; they may also attend tutorials and classes. Few are they who learn probability in private by relying on a textbook as the sole or principal source of inspiration and learning. Although some will say that this book is quite difficult, it is the case that first-year students at many universities learn some quite difficult things, such as axiomatic systems in algebra and ǫ/δ analysis, and we doubt if the material covered here is inherently more challenging than these. Also, lecturers and tutors have certain advantages over authors—they have the power to hear and speak to their audiences—andthese advantagesshould help them explain the harder things to their students. Here are a few words about our approach to rigour. It is impossible to prove everything with complete rigour at this level. On the other hand, we believe it is important that students should understand why rigour is necessary. We try to be rigorous where possible, and else- where we go to some lengths to point out how and where we skate on thin ice. Most sections finish with a few exercises; these are usually completely routine, and students should do them as a matter of course. Each chapter finishes with a collection of problems; these are often much harder than the exercises, and include numerous questions taken from examination papers set in Cambridge and Oxford. We acknowledge permission from Oxford University Press in this regard, and also from the Faculties of Mathematics at Cambridge and Oxford for further questions added in this second edition. There are two use- ful appendices, followed by a final section containing some hints for solving the problems. Problems marked with an asterisk may be more difficult. We hope that the remaining mistakes and misprints are not held against us too much, and that they do not pose overmuch of a hazard to the reader. Only with the kind help of our students have we reduced them to the present level. Finally, we extend our appreciation to our many students in Bristol, Oxford, and Cambridge for their attention, intelligence, and criticisms during our many years of teach- ing probability to undergraduates. GG, DW Cambridge, Oxford February 2014 Contents PART ABASIC PROBABILITY 1 Events and probabilities 3 1.1 Experimentswithchance ....................... 3 1.2 Outcomesandevents.......................... 3 1.3 Probabilities .............................. 6 1.4 Probabilityspaces ........................... 7 1.5 Discretesamplespaces......................... 9 1.6 Conditionalprobabilities. 11 1.7 Independentevents........................... 12 1.8 Thepartitiontheorem ......................... 14 1.9 Probabilitymeasuresarecontinuous. 16 1.10 Workedproblems ........................... 17 1.11 Problems................................ 19 2 Discrete random variables 23 2.1 Probabilitymassfunctions....................... 23 2.2 Examples................................ 26 2.3 Functionsofdiscreterandomvariables . 29 2.4 Expectation............................... 30 2.5 Conditionalexpectationand thepartitiontheorem . 33 2.6 Problems................................ 35 3 Multivariate discrete distributions and independence 38 3.1 Bivariatediscretedistributions. 38 3.2 Expectationinthemultivariatecase . 40 3.3 Independenceofdiscreterandomvariables . 41 3.4 Sumsofrandomvariables ....................... 44 3.5 Indicatorfunctions........................... 45 3.6 Problems................................ 47 4 Probability generating functions 50 4.1 Generatingfunctions.......................... 50 4.2 Integer-valuedrandomvariables. 51 4.3 Moments................................ 54 4.4 Sumsofindependentrandomvariables . 56 4.5 Problems................................ 58 viii Contents 5 Distribution functions and density functions 61 5.1 Distributionfunctions ......................... 61 5.2 Examplesofdistributionfunctions . 64 5.3 Continuousrandomvariables ..................... 65 5.4 Somecommondensityfunctions ................... 68 5.5 Functionsofrandomvariables..................... 71 5.6 Expectationsofcontinuousrandomvariables . 73 5.7 Geometricalprobability ........................ 76 5.8 Problems................................ 79 PART BFURTHER PROBABILITY 6 Multivariate distributions and independence 83 6.1 Randomvectorsandindependence . 83 6.2 Jointdensityfunctions......................... 85 6.3 Marginaldensityfunctionsandindependence . 88 6.4 Sumsofcontinuousrandomvariables . 91 6.5 Changesofvariables.........................

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