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Springer Texts in Statistics
Advisors:
Stephen Fienberg Ingram Olkin
Springer Texts in Statistics
Alfred: Elements of Statistics for the Life and Social Sciences
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Peters: Counting for Something: Statistical Principles and Personalities
Pfeiffer: Probability for Applications
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Continued at end of book
JIM PITMAN
PROBABILITY
. .
184"
Springer
Jim Pitman
Department of Statistics
University of California
Berkeley, CA 94720
Editorial Board
Stephen Fienberg
York University
North York, Ontario N3J 1P3
Canada
Ingram Olkin
Department of Statistics
Stanford University
Stanford, CA 94305 USA
Mathematical Subject Classification (1992): 60-01
Li
ary of Congress Cataloging-in-Publication Data
Pitman, Jim.
Probability / Jim Pitman.
p. cm. -- (Springer texts in statistics)
Includes bibliographical references and index.
ISBN -13: XXXXXXXXXX
1. Probabilities. 1. Title. II. Series.
QA273.P XXXXXXXXXX
519.2--dc XXXXXXXXXX
© 1993 Springer-Verlag New York, Inc.
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Preface
Preface to the Instructor
This is a text for a one-quarter or one-semester course in probability, aimed at stu-
dents who have done a year of calculus. The book is organized so a student can learn
the fundamental ideas of probability from the first three chapters without reliance
on calculus. Later chapters develop these ideas further using calculus tools.
The book contains more than the usual number of examples worked out in detail. It
is not possible to go through all these examples in class. Rather, I suggest that you
deal quickly with the main points of theory, then spend class time on problems from
the exercises, or your own favorite problems. The most valuable thing for students
to learn from a course like this is how to pick up a probability problem in a new
setting and relate it to the standard body of theory. The more they see this happen
in class, and the more they do it themselves in exercises, the better.
The style of the text is deliberately informal. My experience is that students learn
more from intuitive explanations, diagrams, and examples than they do from theo-
ems and proofs. So the emphasis is on problem solving rather than theory.
Order of Topics. The basic rules of probability all appear in Chapter 1. Intuition
for probabilities is developed using Venn and tree diagrams. Only finite additivity of
probability is treated in this chapter. Discussion of countable additivity is postponed
to Section 3.4. Emphasis in Chapter 1 is on the concept of a probability distribution
and elementary applications of the addition and multiplication rules. Combinatorics
appear via study of the binomial and hypergeometric distributions in Chapter 2. The
vi Preface
concepts of mean and standard deviation appear in a preliminary form in this chapter,
motivated by the normal approximation, without the notation of random variables.
These concepts are then developed for discrete random variables in Chapter 3. The
main object of the first three chapters is to get to the circle of ideas around the
normal approximation for sums of independent random variables. This is achieved
y Section 3.3. Sections 3.4 and 3.5 deal with the standard distributions on the non-
negative integers. Conditional distributions and expectations, covariance and co
e-
lation for discrete distributions are postponed to Chapter 6, nea
y treatment of the
same concepts for continuous distributions. The discrete theory could be done right
after Chapter 3, but it seems best to get as quickly as possible to continuous things.
Chapters 4 and 5 treat continuous distributions assuming a calculus background. The
main emphasis here is on how to do probability calculations rather than rigorous
development of the theory. In particular, differential calculations are used freely from
Section 4.1 on, with only occasional discussion of the limits involved.
Optional Sections. These are more demanding mathematically than the main stream
of ideas.
Terminology. Notation and terms are standard, except that outcome space is used
throughout instead of sample space. Elements of an outcome space are called pos-
sible outcomes.
Pace. The earlier chapters are easier than later ones. It is important to get quickly
through Chapters 1 and 2 (no more than three weeks). Chapter 3 is more substantial
and deserves more time. The end of Chapter 3 is the natural time for a midterm
examination. This can be as early as the sixth week. Chapters 4, 5, and 6 take time,
much of it spent teaching calculus.
Preface to the Student
Prerequisites. This book assumes some background of mathematics, in particular,
calculus. A summary of what is taken for granted can be found in Appendices I to
III. Look at these to see if you need to review this material, or perhaps take another
mathematics course before this one.
How to read this book. To get most benefit from the text, work one section at
a time. Start reading each section by skimming lightly over it. Pick out the main
ideas, usually boxed, and see how some of the examples involve these ideas. Then
you may already be able to do some of the first exercises at the end of the section,
which you should try as soon as possible. Expect to go back and forth between the
exercises and the section several times before mastering the material.
Exercises. Except perhaps for the first few exercises in a section, do not expect to
e able to plug into a formula or follow exactly the same steps as an example in the
text. Rather, expect some variation on the main theme, perhaps a combination with
ideas of a previous section, a rea
angement of the formula, or a new setting of the
same principles. Through working problems you gain an active understanding of
Preface vii
the concepts. If you find a problem difficult, or can't see how to start, keep in mind
that it will always be related to material of the section. Try re-reading the section
with the problem in mind. Look for some similarity or connection to get started.
Can you express the problem in a different way? Can you identify relevant variables?
Could you draw a diagram? Could you solve a simpler problem? Could you
eak
up the problem into simpler parts? Most of the problems will yield to this sort of
approach once you have understood the basic ideas of the section. For more on
problem-solving techniques, see the book How to Solve It by G. Polya (Princeton
University Press).
Solutions. Brief solutions to most odd numbered exercises appear at the end of the
ook.
Chapter Summaries. These are at the end of every chapter.
Review Exercises. These come after the summaries at the end of every chapter.
Try these exercises when reviewing for an examination. Many of these exercises
combine material from previous chapters.
Distribution Summaries. These set out the properties of the most important distri-
utions. Familiarity with these properties reduces the amount of calculation required
in many exercises.
Examinations. Some midterm and final examinations from courses taught from this
text are provided, with solutions a few pages later.
Acknowledgments
Thanks to many students and instructors who have read preliminary versions of this
ook and provided valuable feedback. In particular, David Aldous, Peter Bickel, Ed
Chow, Steve Evans, Roman Fresnedo, David Freedman, Alberto Gandolfi, Hank Ib-
ser, Barney Krebs, Bret Larget, Russ Lyons, Lucien Le Cam, Maryse Loranger, Deborah
Nolan, David Pollard, Roger Purves, Joe Romano, Tom Salisbury, David Siegmund,
Anne Sheehy, Philip Stark, and Ruth Williams