The elements of probability (The Addison-Wesley series in behavioral science)

241 Pages·1969·18.87 MB·English

by  Simeon M Berman

THE ELEMENTS OF PROBABILITY

SD1EON M. BERMAN
New York University
THE ELEMENTS OF PROBABILITY
....
TT
ADDISON-WESLEY PUBLISHING COMPANY
Reading, Massachusetts . Menlo Park, California . London' Don Mills, Ontario
The Addison-Wesley Series in
BEHAVIORAL SCIENCE: QUANTITATIVE METHODS
Frederick Mosteller, Consulting Editor
Copyright @ 1969 by Addison-Wesley Publishing Company, Inc. Philippines copyright
1969 by Addison-Wesley Publishing Company, Inc.
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 written permission of the publisher. Printed in
the United States of America. Published simultaneously in Canada. Library of Congress
Catalog Card No. 69-14593.
TO IONA

PREFACE
This is an elementary introduction to the theory and applications of prob
ability. The only formal mathematical knowledge essential for an under
standing of this book is that of the algebraic properties of the real number
system (the "ordered field" properties) as usually presented In high school
algebra. There is no attempt to teach or use calculus here. The only concept
from the latter subject that is used is that of a limit of a sequence of real
numbers; a self-contained exposition is given in Chapter 3.
The aim of the book is the presentation of the more profound, interesting,
and useful results of classical probability in elementary mathematical forms.
The pedagogic principle shaping this book is that important ideas can and
should be taught at the most elementary level: that the student benefits
more from a few profundities than from a lot of tnvia. Cast here in primitive
but meaningful forms are topics which are usually given in more advanced
books: Bernoulli's Law of Large Numbers, the Ghvenko-Cantelli theorem
on the convergence of the empirical distribution function (Clustering
Principle), the random walk, branching processes, Markov chains, and se
quential testing of hypotheses. Applications and numerical illustrations
complement every theoretical result of importance; for example, the coin-
vii
viii PREFACE
tossing model is applied to medical, psychological, and industrial testing
and to electorate polling.
A quick examination of this book will reveal two related features:
1) It is written in the style of plane geometry: definitions and axioms
are stated and their logical consequences are derived. Every result, except
for the indicated few, is rigorously proved. Whenever the complete proof of
a general theorem would require too much notation, we prove only a par
ticular case, in this way simplifying but conserving the ideas of the general
proof.
2) Symbolic notation is minimized: the student is forced to work in
"prose."
The successive chapters strongly depend on their predecessors. The
reader should check every cross-reference to definitions or propositions:
the only way to understand and remember them is to see how they are used.
Exercises follow each section: they consist of simple applications of stated
results, questions about structures of given proofs, and simple theoretical
extensions and verifications. The instructor should preserve the given pro
portion of theoretical to numerical material; for example, he should not
spend most of the class time reviewing the exercises-the expository material
is more important.
The first seven chapters, and the last-more than half the book-are
devoted to the single model of the tossing of a coin n times; no other prob
ability models are introduced until the eighth chapter. The fundamental
notions-random variables, the Law of Large Numbers, normal approxima
tion, events, independence-are first "grown" within the coin-tossing model;
then, only after the reader is at ease in this particular case, they are extended
to the more general case. While most other authors begin with set theory
and the axioms of probability measures, I postpone the "calculus of events"
until there is a natural demand for it-in the random-walk model. Combi
natorial theory is minimized: just enough to derive the binomial distribution
is given. Many other books at this level contain exercises which are applica
tions of elementary combinatorial theory. These have been intentionally
left out of this book: I have found it preferable to develop this systematic
study of the foundations of probability without them.
Another novelty of the exposition is the use of capital letters in nouns
representing random variables and lower-case letters in the same nouns
representing values of the random variables; for example, the Number of
heads in n tosses of a coin refers to a random variable. The terminology used
differs from the conventional at certain points; for example, the traditional
PREFACE IX
"Bernoulli trial" is called just a "coin toss," and Cain gambles with Abel
instead of Peter with Paul.
The material is annotated as follows. Each chapter, with the ell.ception
of Chapter 4, is divided into sections. Definitions, examples, and proposi
tions are labeled by chapter and number; for example, Proposition 7.2 is
the second in Chapter 7. The exercises are numbered only within the section;
each exercise set is numbered. Numbered displays, of which there are not
many. are labeled by chapter and number; they are usually referred to as
"formulas." The tables labeled with roman numerals are located in a sepa
rate section on pages 200-205.
Here is the content of a one-semester course intended for students with
some interest in axiomatic mathematics but not yet well trained in it; they
may have other fields of primary specialization. Even though calculus is
not a prerequisite for this book, it does contain sufficient theoretical material
for those who have had calculus.
I wish to thank Professor Frederick Mosteller for helpful, constructive
comments on the original manuscript; and Professor J. D. Kuelbs for
additional remarks. I appreciate the permission of Holden-Day. Inc., to
use Tables I and III.
September 1968 S. M. B.
Brooklyn, New York