Elementary Probability Theory with Stochastic Processes

331 Pages·1975·14.284 MB·English

by  Kai Lai Chung (auth.)

Undergraduate Texts in Mathematics
Editors: F. W. Gehring
P. R. Halmos ·
Advisory Board: C. DePrima
I. Herstein
J. Kiefer
W. LeVeque
Kai Lai Chung
Elementary
Probability Theory
with Stochastic Processes
Springer Science+Business Media, LLC
Kai Lai Chung
Professor of Mathematics
Stanford University
Department of Mathematics
Stanford, California 94305
AMS Subject Classification ( 197 4)
Primary: 6001
Secondary: 60C05, 60110, 60115, 60J20, 60J75
Library of Congress Cataloging in Publication Data
Chung, Kai Lai, 1917-
Elementary probability theory with stochastic
processes.
(Undergraduate texts in mathematics)
Bibliography: p.
Includes index.
1. Probabilities. 2. Stochastic processes.
I. Title.
QA273.C5774 1975 519.2 75-30544
All rights reserved.
No part of this book may be translated or reproduced in
any form without written permission from
Springer Science+ Business Media, LLC.
© 1974 , 1975 by Springer Science+B usiness Media New York
Originally published by Springer-Verlag New York in 1975
Softcover reprint of the hardcover 2nd edition 1975
Second Printing
ISBN 978-1-4757-5116-1 ISBN 978-1-4757-5114-7 (eBook)
DOI 10.1007/978-1-4757-5114-7
Preface to the First Edition
In the past half-century the theory of probability has grown from a minor
isolated theme into a broad and intensive discipline interacting with many
other branches of mathematics. At the same time it is playing a central role
in the mathematization of various applied sciences such as statistics, opera
tions research, biology, economics and psychology-to name a few to which
the prefix "mathematical" has so far been firmly attached. The coming-of-age
of probability has been reflected in the change of contents of textbooks on
the subject. In the old days most of these books showed a visible split
personality torn between the combinatorial games of chance and the so-called
"theory of errors" centering in the normal distribution. This period ended
with the appearance of Feller's classic treatise (see [Feller l]t) in 1950, from
the manuscript of which I gave my first substantial course in probability.
With the passage of time probability theory and its applications have won a
place in the college curriculum as a mathematical discipline essential to many
fields of study. The elements of the theory are now given at different levels,
sometimes even before calculus. The present textbook is intended for a course
at about the sophomore level. It presupposes no prior acquaintance with the
subject and the first three chapters can be read largely without the benefit of
calculus. The next three chapters require a working knowledge of infinite
series and reiated topics, and for the discussion involving random variables
with densities some calculus is of course assumed. These parts dealing with
the "continuous case" as distinguished from the "discrete case" are easily
separated and may be postponed. The contents of the first six chapters should
form the backbone of any meaningful first introduction to probability theory.
Thereafter a reasonable selection includes: §7.1 (Poisson distribution, which
may be inserted earlier in the course), some kind of going over of §7.3, 7.4, 7.6
(normal distribution and the law oflarge numbers), and §8.1 (simple random
walks which are both stimulating and useful). All this can be covered in a
semester but for a quarter system some abridgment will be necessary. Specifi
cally, for such a short course Chapters 1 and 3 may be skimmed through and
the asterisked material omitted. In any case a solid treatment of the normal
approximation theorem in Chapter 7 should be attempted only if time is
available as in a semester or two-quarter course. The final Chapter 8 gives a
self-contained elementary account of Markov chains and is an extension of
the main course at a somewhat more mature level. Together with the aster
isked sections 5.3, 5.4 (sequential sampling and Polya urn scheme) and 7.2
(Poisson process), and perhaps some filling in from the Appendices, the
material provides a gradual and concrete passage into the domain of sto-
t Names in square brackets refer to the list of General References on p. 307. William Feller
(1906-1970).
vi Preface to the First Edition
chastic processes. With these topics included the book will be suitable for a
two-quarter course, of the kind that I have repeatedly given to students of
mathematical sciences and engineering. However, after the preparation of
the first six chapters the reader may proceed to more specialized topics treated
e.g. in the above mentioned treatise by Feller. If the reader has the adequate
mathematical background, he will also be prepared to take a formal rigorous
course such as presented in my own more advanced book [Chung 1].
Much thought has gone into the selection, organization and presentation
of the material to adapt it to classroom uses, but I have not tried to offer a
slick package to fit in with an exact schedule or program such as popularly
demanded at the quick-service counters. A certain amount of flexibility and
choice is left to the instructor wh0 can best judge what is right for his class.
Each chapter contains some easy reading at the beginning, for motivation
and illustration, so that the instructor may concentrate on the more formal
aspects of the text. Each chapter also contains some slightly more challenging
topics (e.g., §1.4, 2.5) for optional sampling. They are not meant to deter the
beginner but to serve as an invitation to further study. The prevailing empha
sis is on the thorough and deliberate discussion of the basic concepts and
techniques of elementary probability theory with few frills and minimal tech
nical complications. Many examples are chosen to anticipate the beginners'
difficulties and to provoke better thinking. Often this is done by posing and
answering some leading questions. Historical, philosophical and personal
comments are inserted to add flavor to this lively subject. It is my hope that
the reader will not only learn something from the book but may also derive
a measure of enjoyment in so doing.
There are over two hundred exercises for the first six chapters and some
eighty more for the last two. Many are easy, the harder ones indicated by
asterisks, and all answers gathered at the end of the book. Asterisked sections
and paragraphs deal with more special or elaborate material and may be
skipped, but a little browsing in them is recommended.
The author of any elementary textbook owes of course a large debt to
innumerable predecessors. More personal indebtedness is acknowledged be
low. Michel Nadzela wrote up a set of notes for a course I gave at Stanford
in 1970. Gian-Carlo Rota, upon seeing these notes, gave me an early impetus
toward transforming them into a book. D. G. Kendall commented on the
first draft of several chapters and lent further moral support. J. L. Doob
volunteered to read through most of the manuscript and offered many helpful
suggestions. K. B. Erickson used some of the material in a course he taught.
A. A. Balkema checked the almost final version and made numerous improve
ments. Dan Rudolph read the proofs together with me. Perfecto Mary drew
those delightful pictures. Gail Lemmond did the typing with her usual ef
ficiency and dependability. Finally, it is a pleasure to thank my old publisher
Springer-Verlag for taking my new book to begin a new series of under
graduate texts.
K. L. C.
March 1974.
Preface to the Second Edition
A determined effort was made to correct the errors in the first edition.
This task was assisted by: Chao Hung-po, J. L. Doob, R. M. Exner, W. H.
Fleming, A.M. Gleason, Karen Kafador, S. H. Polit, and P. van Moerbeke.
Miss Kafador and Dr. Polit compiled particularly careful lists of suggestions.
The most distressing errors were in the Solutions to Problems. All of them
have now been checked by myself from Chapter 1 to 5, and by Mr. Chao
from Chapter 6 to 8. It is my fervent hope that few remnant mistakes remain
in that sector. A few small improvements and additions were also made, but
not all advice can be heeded at this juncture. Users of the book are implored
to send in any criticism and commentary, to be taken into consideration in a
future edition. Thanks are due to the staff of Springer-Verlag for making this
revision possible so soon after the publication of the book.
K. L. C.
CONTENTS
Preface
Chapter 1: Set 1
1.1 Sample sets I
1.2 Operations with sets 3
1.3 Various relations 6
1.4 Indicator 12
Exercises 16
Chapter 2: Probability 18
2.1 Examples of probability 18
2.2 Definition and illustrations 21
2.3 Deductions from the axioms 28
2.4 Independent events 31
2.5 Arithmetical density 36
Exercises 39
Chapter 3: Counting 43
3.1 Fundamental rule 43
3.2 Diverse ways of sampling 46
3.3 Allocation models; binomial coefficients 52
3.4 How to solve it 59
Exercises 66
Chapter 4: Random Variables 71
4.1 What is a random variable? 71
4.2 How do random variables come about? 75
4.3 Distribution and expectation 80
4.4 Integer-valued random variables 86
4.5 Random variables with densities 90
4.6 General case 100
Exercises 104
Appendix 1 : Borel Fields and General Random Variables 109
Chapter 5 : Conditioning and Independence 111
5.1 Examples of conditioning Ill
5.2 Basic formulas 116
Contents
X
5.3 Sequential sampling 125
5.4 Polya's urn scheme 129
5.5 Independence and relevance 134
5.6 Genetical models 144
Exercises 150
Chapter 6: Mean, Variance and Transforms 156
6.1 Basic properties of expectation 156
6.2 The density case 160
6.3 Multiplication theorem; variance and covariance 164
6.4 Multinomial distribution 171
6.5 Generating function and the like 177
Exercises 185
Chapter 7: Poisson and Normal Distributions 192
7.1 Models for Poisson distribution 192
7.2 Poisson process 199
7.3 From binomial to normal 210
7.4 Normal distribution 217
7.5 Central limit theorem 220
7.6 Law of large numbers 227
Exercises
Appendix 2: Stirling's Formula and DeMoivre-Laplace's Theorem 237
Chapter 8: From Random Walks to Markov Chains 240
8.1 Problems of the wanderer or gambler 240
8.2 Limiting $Chemes 246
8.3 Transition probabilities 252
8.4 Basic structure of Markov chains 260
8.5 Further developments 267
8.6 Steady state 274
8.7 Winding up (or down?) 286
Exercises 296
Appendix 3: Martingale 305
General References 306
Answers to Problems 309
Index 323
Chapter 1
Set
1.1. Sample sets
These days school children are taught about sets. A second grader* was
asked to name "the set of girls in his class." This can be done by a complete
list such as:
"Nancy, Florence, Sally, Judy, Ann, Barbara, "
A problem arises when there are duplicates. To distinguish between two
Barbaras one must indicate their family names or call them B1 and B2. The
same member cannot be counted twice in a set.
The notion of a set is common in all mathematics. For instance in geom
etry one talks about "the set of points which are equi-distant from a given
point." This is called a circle. In algebra one talks about "the set of integers
which have no other divisors except l and itself." This is called the set of
prime numbers. In calculus the domain of definition of a function is a set of
numbers, e.g., the interval (a, b); so is the range of a function if you remember
what it means.
In probability theory the notion of a set plays a more fundamental role.
Furthermore we are interested in very general kinds of sets as well as specific
concrete ones. To begin with the latter kind, consider the following examples:
(a) a bushel of apples;
(b) fifty five cancer patients under a certain medical treatment;
(c) all the students in a college;
(d) all the oxygen molecules i:Q. a given container;
(e) all possible outcomes when six dice are rolled;
(f) all points on a target board.
Let us consider at the same time the following "smaller" sets:
(a') the rotten apples in that bushel;
(b') those patients who respond positively to the treatment;
(c') the mathematics majors of that college;
(d ') those molecules which are traveling upwards;
(e ') those cases when the six dice show different faces;
(f') the points in a little area called the "hull's eye" on the board.
* My son Daniel.
1
2 Set
We shall set up a mathematical model for these and many more such
examples that may come to mind, namely we shall abstract and generalize
our intuitive notion of "a bunch of things." First we call the things points,
then we call the bunch a space; we prefix them by the word "sample" to
distinguish these terms from other usages, and also to allude to their statistical
origin. Thus a sample point is the abstraction of an apple, a cancer patient,
a student, a molecule,, a possible chance outcome, or an ordinary geometrical
point. The sample space consists of a number of sample points, and is just a
name for the totality or aggregate of them all. Any one of the examples (a)-(f)
above can be taken to be a sample space, but so also may any one of the
smaller sets in (a ')-(f'). What we choose to call a space [a universe] is a relative
matter.
Let us then fix a sample space to be denoted by !J, the capital Greek letter
omega. It may contain any number of points, possibly infinite but at least one.
(As you have probably found out before, mathematics can be very pedantic!)
Any of these points may be denoted by w, the small Greek letter omega, to
be distinguished from one another by various devices such as adding sub
scripts or dashes (as in the case of the two Barbaras if we do not know their
family names), thus wr, w2, w', .... Any partial collection of the points is a
subset of!J, and since we have fixed Q we will just call it a set. In extreme cases
a set may be Q itself or the empty set which has no point in it. You may be
surprised to hear that the empty set is an important entity and is given a
special symbol 0. The number of points in a set S will be called its size and
denoted by IS f, thus it is a nonnegative integer or oo. In particular 101 = 0.
A particular set Sis well defined if it is possible to tell whether any given
point belongs to it or not. These two cases are denoted respectively by
w E::: S; w e_ S.
Thus a set is determined by a specified rule of membership. For instance, the
sets in (a')-(f') are well defined up to the limitations of verbal descriptions.
One can always quibble about the meaning of words such as "a rotten apple,"
or attempt to be funny by observing, for instance, that when dice are rolled
on a pavement some of them may disappear into the sewer. Some people of
a pseudo-philosophical turn of mind get a lot of mileage out of such caveats,
but we will not indulge in them here. Now, one sure way of specifying a rule
to determine a set is to enumerate all its members, namely to make a complete
list as the second grader did. But this may be tedious if not impossible. For
example, it will be shown in §3.1 that the size of the set in (e) is equal to
66 = 46656. Can you give a quick guess how many pages of a book like this
will be needed just to record all these possibilities of a mere throw of six dice?
On the other hand it can be described in a systematic and unmistakable way
as the set of all ordered 6-tuples of the form below: