The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)

482 Pages·1996·28.244 MB·English

by  Carol Ash

CAROL ASH
THE
PROBABILITY
TUTORING
BOOK A
A.
Revised Printing
AN
INTUITIVE
COURSE
FOR ENGINEERS
AND SCIENTISTS
(and everyone else!)

The Probability Tutoring Book
AN INTUITIVE COURSE
FOR ENGINEERS AND SCIENTISTS
(AND EVERYONE ELSE!)
Revised Printing
Carol Ash
University of Illinois
at Urbana-Champaign
IEEE
PRESS
The Institute of Electrical and Electronics Engineers, Inc., New York
IEEE PRESS
PO Box 1331,445 Hoes Lane
Piscataway, New Jersey 08855-1331
1992 PRESS Editorial Board
William Perkins, Editor in Chief
К. K. Agarwal K. Hess A. C. Schell
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D. M. Etter E. K. Miller Y. Sunahara
J. J. Farrell III J. M. F. Moura J. W. Woods
J. G. Nagle
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Anne Reifsnyder, Associate Editor
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©1993 by the Institute of Electrical and Electronics Engineers, Inc.
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Printed in the United States of America
10 9 8 7 6
ISBN 0-7803-1051-9 [pbk]
IEEE Order Number: PP0288-1 [pbk]
ISBN 0-87942-293-0
IEEE Order Number: PC0288-1
Library of Congress Cataloging-in-Publication Data
Ash, Carol (date)
The probability tutoring book : an intuitive course for engineers
and scientists (and everyone else!) / Carol Ash.
p. cm.
“IEEE order number: PC0288-l”-T.p. verso.
Includes index.
ISBN 0-87942-293-9
1. Engineering mathematics. 2. Probabilities. I. Title.
TA340.A75 1993
62O’.OO1’51—dc20 92-53183
CIP
Contents
Preface vii
Introduction ix
DISCRETE PROBABILITY
CHAPTER 1 Basic Probability 1
1-1 Probability Spaces 1
1-2 Counting Techniques 6
1-3 Counting Tbchniques Continued 14
1-4 ORs and AT LEAST? 20
1-5 Continuous Uniform Distributions 30
Review for the Next Chapter 36
CHAPTER 2 Independent Trials and 2-Stage Experiments 37
2-1 Conditional Probability and Independent Events 37
2-2 The Binomial and Multinomial Distributions 46
2-3 Simulating an Experiment 53
2-4 The Theorem of Total Probability and Bayes’ Theorem 56
2-5 The Poisson Distribution 63
Review Problems for Chapters 1 and 2 68
Iv Contents
CHAPTER 3 Expectation 74
3-1 Expected Value of a Random Variable 74
3-2 The Method of Indicators 77
3-3 Conditional Expectation and the Theorem of Total Expectation 85
3-4 Variance 91
Review Problems for Chapter 3 92
Review for the Next Chapter 95
CONTINUOUS PROBABILITY
CHAPTER 4 Continuous Random Variables 97
4-1 Density Functions 97
4-2 Distribution Functions 104
4-3 The Exponential Distribution 120
4-4 The Normal (Gaussian) Distribution 127
4-5 The Election Problem 138
4-6 Functions of a Random Variable 141
4-7 Simulating a Random Variable 158
Review for the Next Chapter 164
CHAPTERS J ointly Distributed Random Variables 171
5-1 Joint Densities 171
5-2 Marginal Densities 182
5-3 Functions of Several Random Variables 192
Review Problems for Chapters 4 and 5 197
CHAPTER 6 Jointly Distributed Random Variables Continued 201
6-1 Sums of Independent Random Variables 201
6-2 Order Statistics 207
CHAPTER 7 Expectation Again 213
7-1 Expectation of a Random Variable 213
7-2 Variance 223
7-3 Correlation 235
CHAPTER 8 Conditional Probability 242
8-1 Conditional Densities 242
8-2 2-Stage Experiments 250
Contents v
8-3 Mixed 2-Stage Experiments 257
Review Problems for Chapters 6, 7, and 8 263
CHAPTER 9 Limit Theorems 266
9-1 The Central Limit Theorem 266
9-2 The Weak Law of Large Numbers 269
Review Problems for Chapters 4-9 274
SOLUTIONS 275
INDEX 466

Preface
This is the second in a series of tutoring books,1 a text in probability, written
for students in mathematics and applied areas such as engineering, physics,
chemistry, economics, computer science, and statistics. The style is unlike that
of the usual mathematics text, and I’d like to describe the approach and explain
the rationale behind it.
Mathematicians and consumers of mathematics (such as engineers) seem
to disagree as to what mathematics actually is. To a mathematician, it’s im­
portant to distinguish between rigor and informal thinking. To an engineer,
intuitive thinking, geometric reasoning, and physical argument are all valid if
they illuminate a problem, and a formal proof is often unnecessary or coun­
terproductive.
The typical mathematics text includes applications and examples, but its
dominant feature is formalism. Theorems and definitions are stated precisely,
and many results are proved at a level of rigor that is acceptable to a working
mathematician. This is bad. After teaching many undergraduates, most quite
competent and some, in fact, blindingly bright, it seems entirely clear to me
that most are not ready for an abstract presentation. At best, they will have
a classroom teacher who can translate the formalism into ordinary English
(“what this really means is ... ”). At worst, they will give up. Most will simply
learn to read around the abstractions so that the textbook at least becomes
useful as a source of examples.
This text uses informal language and thinking whenever possible. This is
the appropriate approach even for mathematics majors: Rigorous probability
1The first is The Calculus Tutoring Book by Carol Ash and Robert Ash (New York: IEEE
Press, 1986).
vii
vill Preface
isn’t even possible until you’ve had a graduate-level course in measure theory,
and it isn’t meaningful until you’ve had this informal version first.
In any textbook, problems are as important to the learning process as
the text material itself. I chose the problems in this book carefully, and much
consideration was given to the number of problems, so that if you do most of
them you will get a good workout. To be of maximum benefit to students, the
text includes detailed solutions (prepared by the author) to all problems.
I’d like to thank the staff at the IEEE PRESS, Dudley Kay, Executive Ed­
itor; Denise Gannon, Production Supervisor; and Anne Reifsnyder, Associate
Editor. I appreciate the time and energy spent by the reviewers, Dr. Robert
E. Lover and members of the IEEE Press Board.
Most of all I owe my husband, Robert B. Ash, for patiently and critically
reading every word and for being better than anyone else at teaching mathe­
matics in general, and probability in particular, to students and wives.
Carol Ash