Collection of problems in probability theory

Collection of problems in probability theory

L. D. MESHALKIN Moscow State University Collection of problems in probability theory Translated from the Russian and edited by LEO F. BORON University of Idaho and BRYAN A. HAWORTH University of Idaho and California State College, Bakersfield NOORDHOFF INTERNATIONAL PUBLISHING, LEYDEN

1973 Noordhoff International Publishing, Leyden, The Netherlands Softcover reprint of the hardcover 1st edition 1973 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, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN -13: 978-94-010-2360-3 001: 10.1007/978-94-010-2358-0 e-isbn -13: 978-94-010-2358-0 Library of Congress Catalog Card Number: 72-76789 Original title "Sbornik zadach po teorii veroyatnostey" published in 1963 in Moscow

Contents Editor's foreword Foreword to the Russian edition 1 Fundamental concepts viii ix 1.1 Field of events 3 1.2 Interrelationships among cardinalities of sets 5 1.3 Definition of probability 7 1.4 Classical definition of probability. Combinatorics 8 1.5 Simplest problems on arrangements 11 1.6 Geometric probability 13 1.7 Metrization and ordering of sets 15 2 Application of the basic formu]as 17 2.1 Conditional probability. Independence 20 2.2 Discrete distributions: binomial, multinomial, geometric, hypergeometric 23 2.3 Continuous distributions 27 2.4 Application of the formula for total probability 29 2.5 The probability of the sum of events 31 2.6 Setting up equations with the aid of the formula for total probability 32 3 Random variables and their properties 35 3.1 Calculation of mathematical expectations and dispersion 39 v

Contents 3.2 Distribution functions 3.3 Correlation coefficient 3.4 Chebyshev's inequality 3.5 Distribution functions of random variables 3.6 Entropy and information 44 45 46 48 52 4 Basic limit theorems 56 4.1 The de Moivre-Laplace and Poisson theorems 58 4.2 Law of Large Numbers and convergence in probability 63 4.3 Central Limit Theorem 66 5 Characteristic and generating functions 71 5.1 Calculation of characteristic and generating functions 5.2 Connection with properties of a distribution 5.3 Use of the c.f. and g.f. to prove the limit theorems 5.4 Properties of c.f.'s and g.f.'s 5.5 Solution of problems with the aid of c.f.'s and g.f.'s 72 73 76 78 79 6 Application of measure theory 82 6.1 Measurability 85 6.2 Various concepts of convergence 86 6.3 Series of independent random variables 87 6.4 Strong law of large numbers and the iterated logarithm law 89 6.5 Conditional probabilities and conditional mathematical expectations 93 vi

7 Infinitely divisible distributions. Normal law. Multidimensional distributions Contents 96 7.1 Infinitely divisible distributions 7.2 The normal distribution 7.3 Multidimensional distributions 96 100 103 8 Markov chains 107 8.1 Definition and examples. Transition probability matrix 108 8.2 Classification of states. Ergodicity 112 8.3 The distribution of random variables defined on a Markov chain 115 Appendix 117 Answers 125 Suggested reading 145 Index 147 vii

Editor's foreword The Russian version of A collection of problems in probability theory contains a chapter devoted to statistics. That chapter has been omitted in this translation because, in the opinion of the editor, its content deviates somewhat from that which is suggested by the title: problems in probability theory. The original Russian version contains some errors; an attempt was made to correct all errors found, but perhaps a few stiii remain. An index has been added for the convenience of the reader who may be searching for a definition, a classical problem, or whatever. The index lists pages as well as problems where the indexed words appear. The book has been translated and edited with the hope of leaving as much "Russian flavor" in the text and problems as possible. Any peculiarities present are most likely a result of this intention. August, 1972 Bryan A. Haworth viii

Foreword to the Russian edition This Collection of problems in probability theory is primarily intended for university students in physics and mathematics departments. Its goal is to help the student of probability theory to master the theory more profoundly and to acquaint him with the application of probability theory methods to the solution of practical problems. This collection is geared basically to the third edition of the GNEDENKO textbook Course in probability theory, Fizmatgiz, Moscow (1961), Probability theory, Chelsea (1965). It contains 500 problems, some suggested by monograph and journal article material, and some adapted from existing problem books and textbooks. The problems are combined in nine chapters which are equipped with short introductions and subdivided in turn into individual sections. The problems of Chapters 1-4 and part of 5,8 and 9 correspond to the semester course Probability theory given in the mechanics and mathematics department of MSU. The problems of Chapters 5-8 correspond to the semester course Supplementary topics in probability theory. Difficult problems are marked with an asterisk and are provided with hints. Several tables are adjoined to the collection. Answers are given only to odd numbered problems. This is done to train the student to evaluate independently the correctness of a solution, and also so that the material of the collection could be used for supervised work. To supplement the collection, the teacher can make use of the following three problem books which contain well chosen material on statistics and the theory of stochastic processes: 1. VOLODIN, B. G., M. P. GANIN, I. YA. DINER, L. B. KOMAROV, A. A. SVESHNIKOV, and K. B. STAROBIN. Textbook on problem solving ix

in probability theory for engineers. Sudpromgiz, Leningrad (1962). 2. LAJOS TAKAcS. Stochastic processes. Problems and solutions. Wiley (1970) (in the series Methuen's monographs on applied probability and statistics). 3. DAVID, F. N. and E. S. PEARSON. Elementary statistical exercises. Cambridge University Press (1961). My co-workers and degree candidates of the MSU Department of Probability Theory were of enormous help in choosing and formulating these exercises. I am deeply indebted to them for this. In particular I wish to thank M. Arato, B. V. Gnedenko, R. L. Dobrushin and Ya. G. Sinai. July 9, 1963 L. D. Meshalkin x