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PROBABILITY AND STATISTICAL INFERENCE
Tenth Edition
ROBERT V. H OGG
ELLIOT A. TANIS
DALE L. ZIMMERMAN
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Library of Congress Cataloging-in-Publication Data
Names: Hogg, Robert V., author. | Tanis, Elliot A., author. | Zimmerman, Dale
L., author.
Title: Probability and statistical inference / Robert V. Hogg, Elliot
A. Tanis, Dale L. Zimmerman.
Description: Tenth edition. | New York, NY : Pearson, 2018. | Includes
bibliographical references and index.
Identiers: LCCN 2018016546 | ISBN 9780135189399 | ISBN 013518939X
Subjects: LCSH: Probabilities. | Mathematical statistics.
Classication: LCC QA273 .H694 2018 | DDC 519.5–dc23 LC record available at
https://urldefense.proofpoint.com/v2/url? u=https-3A__lccn.loc.gov_2018016546&d=DwIFAg&c=
0YLnzTkWOdJlub_y7qAx8Q&r=wPGJMKK17CeHqRrBmHL3dANcU7vAP6xMwR_
eHf4JPBM&m=grcVj3gROiayls_YZE61JriBkuhmiKnZjntz0t8EFQ4&s=OPgbnDS6cllWkFLqGuCc-
CdBCX8__PFmMdvdQp5zyZw&e=
118
www.pearsonhighered.com
ISBN-13: 978-0-13-518939-9
ISBN-10: 0-13-518939-X
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CONTENTS
Preface v
Prologue ix
1
PROBABILITY 1
1.1 Properties of Probability 1
1.2 Methods of Enumeration 11
1.3 Conditional Probability 20
1.4 Independent Events 29
1.5 Bayes’ Theorem 35
2
DISCRETE DISTRIBUTIONS 41
2.1 Random Variables of the Discrete Type 41
2.2 Mathematical Expectation 47
2.3 Special Mathematical Expectations 53
2.4 The Binomial Distribution 63
2.5 The Hypergeometric Distribution 71
2.6 The Negative Binomial Distribution 76
2.7 The Poisson Distribution 81
3
CONTINUOUS DISTRIBUTIONS 91
3.1 Random Variables of the Continuous
Type
91
3.2 The Exponential, Gamma, and Chi-Square
Distributions
100
3.3 The Normal Distribution 110
3.4 Additional Models 119
4
BIVARIATE DISTRIBUTIONS 129
4.1 Bivariate Distributions of the Discrete Type
129
4.2 The Correlation Coefficient 139
4.3 Conditional Distributions 145
4.4 Bivariate Distributions of the Continuous
Type
153
4.5 The Bivariate Normal Distribution 162
5
DISTRIBUTIONS OF FUNCTIONS
OF
RANDOM VARIABLES 169
5.1 Functions of One Random Variable 169
5.2 Transformations of Two Random Variables
178
5.3 Several Independent Random Variables 187
5.4 The Moment-Generating Function Technique
194
5.5 Random Functions Associated with Normal
Distributions
199
5.6 The Central Limit Theorem 207
5.7 Approximations for Discrete Distributions
213
5.8 Chebyshev’s Inequality and Convergence in
Probability
220
5.9 Limiting Moment-Generating Functions 224
6
POINT ESTIMATION 233
6.1 Descriptive Statistics 233
6.2 Exploratory Data Analysis 245
6.3 Order Statistics 256
6.4 Maximum Likelihood and Method of
Moments Estimation
264
6.5 A Simple Regression Problem 277
6.6 Asymptotic Distributions of Maximum
Likelihood Estimators
285
6.7 Sufficient Statistics 290
6.8 Bayesian Estimation 298
iii
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iv Contents
7
INTERVAL ESTIMATION 307
7.1 Confidence Intervals for Means 307
7.2 Confidence Intervals for the Difference
of Two Means
314
7.3 Confidence Intervals for Proportions 323
7.4 Sample Size 329
7.5 Distribution-Free Confidence Intervals
for Percentiles
337
7.6 More Regression 344
7.7 Resampling Methods 353
8
TESTS OF STATISTICAL
HYPOTHESES 361
8.1 Tests About One Mean 361
8.2 Tests of the Equality of Two Means 369
8.3 Tests for Variances 378
8.4 Tests About Proportions 385
8.5 Some Distribution-Free Tests 392
8.6 Power of a Statistical Test 403
8.7 Best Critical Regions 410
8.8 Likelihood Ratio Tests 418
9
MORE TESTS 425
9.1 Chi-Square Goodness-of-Fit Tests 425
9.2 Contingency Tables 435
9.3 One-Factor Analysis of Variance 446
9.4 Two-Way Analysis of Variance 456
9.5 General Factorial and 2
k
Factorial Designs
465
9.6 Tests Concerning Regression and Correlation
471
9.7 Statistical Quality Control 477
APPENDICES
A
REFERENCES 489
B
TABLES 491
C
ANSWERS TO ODD-NUMBERED
EXERCISES 513
D
REVIEW OF SELECTED
MATHEMATICAL TECHNIQUES 525
D.1 Algebra of Sets
525
D.2 Mathematical Tools for the Hypergeometric
Distribution
529
D.3 Limits 532
D.4 Infinite Series 533
D.5 Integration 537
D.6 Multivariate Calculus 539
Index 545
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PREFACE
In this Tenth Edition of Probability and Statistical Inference, Elliot Tanis and Dale
Zimmerman would like to acknowledge the many contributions that Robert Hogg
made to the rst nine editions. Dr. Hogg died on December 23, 2014, but his insights
continue on in this tenth edition. We are indebted to his inuence on our lives and
work.
CONTENT AND COURSE PLANNING
This text is designed for a two-semester course, but it can be adapted for a one-
semester course. A good calculus background is needed, but no previous study of
probability or statistics is required.
This new edition has more than 25 new examples and more than 75 new exer-
cises. However, its chapters are organized in much the same manner as in the
ninth edition. The rst ve again focus on probability, including the following
topics: conditional probability, independence, Bayes’ theorem, discrete and contin-
uous distributions, certain mathematical expectations including moment-generating
functions, bivariate distributions along with marginal and conditional distributions,
correlation, functions of random variables and their distributions, the central limit
theorem, and Chebyshev’s inequality. We added a section on the hypergeomet-
ric distribution, adding to material that had previously been scattered throughout
the rst and second chapters. Also, to this portion of the book we added material
on new topics, including the index of skewness and the laws of total probability
for expectations and the variance. While the strong probability coverage of the
rst ve chapters is important for all students, feedback we have received indi-
cates that it has been particularly helpful to actuarial students who are studying for
Exam P in the Society of Actuaries’ series (or Exam 1 of the Casualty Actuarial
Society).
The remaining four chapters of the book focus on statistical inference. Topics
carried over from the previous edition include descriptive and order statistics,
point estimation including maximum likelihood and method of moments estima-
tion, sufcient statistics, Bayesian estimation, simple linear regression, interval
estimation, and hypothesis testing. New material has been added on the top-
ics of percentile matching and the invariance of maximum likelihood estima-
tion, and we’ve added a new section on hypothesis testing for variances, which
also includes condence intervals for a variance and for the ratio of two vari-
ances. We present condence intervals for means, variances, proportions, and
regression coefcients; distribution-free condence intervals for percentiles; and
resampling methods (in particular, bootstrapping). Our coverage of hypothesis
testing includes standard tests on means (including distribution-free tests), vari-
ances, proportions, and regression coefcients, power and sample size, best critical
regions (Neyman-Pearson), and likelihood ratio tests. On the more applied side,
we describe chi-square tests for goodness of t and for association in contingency
tables, analysis of variance including general factorial designs, and statistical quality
control.
v
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vi Preface
The rst semester of the course should contain most of the topics in Chapters 1–5.
The second semester includes some topics omitted there and many of those in
Chapters 6–9. A more basic course might omit some of the starred sections, but
we believe that the order of topics will give the instructor the exibility needed in
his or her course. The usual nonparametric and Bayesian techniques are placed at
appropriate places in the text rather than in separate chapters. We nd that many
persons like the applications associated with statistical quality control in the last
section.
The Prologue suggests many elds in which statistical methods can be used. At
the end of each chapter, we give some interesting historical comments, which have
proven to be very worthwhile in the past editions. The answers given in this text
for exercises that involve the standard distributions were often calculated using our
probability tables which, of course, are rounded off for printing. If you use a statistical
package, your answers may differ slightly from those given.
ANCILLARIES
Data sets for this text are available on Pearson’s Student Resources website:
https://www.pearson.com/math-stats-resources.
An Instructor’s Solutions Manual containing worked-out solutions to the even-
numbered exercises in the text is available for download from Pearson Education
Instructor’s Resource website: https://www.pearson.com/us/sign-in.html.
Some of the numerical exercises were solved with Maple. For additional
exercises that involve simulations, a separate manual, Probability & Statistics:
Explorations with MAPLE, second edition, by Zaven Karian and Elliot Tanis, is
available for download from Pearson’s Student Resources website. This is located
at https://www.pearson.com/math-stats-resources. Several exercises in that manual
also make use of the power of Maple as a computer algebra system.
so that they can be corrected in a future printing. These errata will also be posted on
http://homepage.divms.uiowa.edu/∼dzimmer/.
ACKNOWLEDGMENTS
We wish to thank our colleagues, students, and friends for many suggestions and
for their generosity in supplying data for exercises and examples. In particular we
would like thank the reviewers of the ninth edition who made suggestions for this
edition. They are Maureen Cox from St. Bonaventure University, Lynne Seymour
from the University of Georgia, Kevin Keen from the University of Northern British
Columbia, Clifford Taylor from Concordia University Ann Arbor, Melanie Autin
from Western Kentucky University, Aubie Anisef from Douglas College, Manohar
Aggarwal from the University of Memphis, Joseph Huber from the University of
Kansas, and Christopher Swanson from Ashland University. Mark Mills from Central
College in Iowa, Matthew Bognar from the University of Iowa, and David Schweitzer
from Liberty University also made many helpful comments, and Hongda Zhang
of the University of Iowa wrote solutions to some of the new exercises. We also
acknowledge the excellent suggestions from our copy editor Jody Callahan and the
ne work of our accuracy checker Kyle Siegrist. We also thank the University of
Iowa and Hope College for providing ofce space and encouragement. Finally, our
families have been most understanding during the preparation of all of this material.
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Preface vii
We would especially like to thank our wives, Elaine and Bridget. We truly appreciate
their patience and needed their love.
Elliot A. Tanis
tanis@hope.edu
Dale L. Zimmerman
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PROLOGUE
The discipline of statistics deals with the collection and analysis of data. Advances
in computing technology, particularly in relation to changes in science and business,
have increased the need for more statistical scientists to examine the huge amount
of data being collected. We know that data are not equivalent to information. Once
data (hopefully of high quality) are collected, there is a strong need for statisticians to
make sense of them. That is, data must be analyzed in order to provide information
upon which decisions can be made. In light of this great demand, opportunities for
the discipline of statistics have never been greater, and there is a special need for
more bright young persons to go into statistical science.
If we think of elds in which data play a major part, the list is almost endless:
accounting, actuarial science, atmospheric science, biological science, economics,
educational measurement, environmental science, epidemiology, nance, genetics,
manufacturing, marketing, medicine, pharmaceutical industries, psychology, sociol-
ogy, sports, and on and on. Because statistics is useful in all these areas, it really should
be taught as an applied science. Nevertheless, to go very far in such an applied science,
it is necessary to understand the importance of creating models for each situation
under study. Now, no model is ever exactly right, but some are extremely useful as an
approximation to the real situation. To be applied properly, most appropriate models
in statistics require a certain mathematical background in probability. Accordingly,
while alluding to applications in the examples and exercises, this textbook is really
about the mathematics needed for the appreciation of probabilistic models necessary
for statistical inferences.
In a sense, statistical techniques are really the heart of the scientic method.
Observations are made that suggest conjectures. These conjectures are tested, and
data are collected and analyzed, providing information about the truth of the conjec-
tures. Sometimes the conjectures are supported by the data, but often the conjectures
need to be modied and more data must be collected to test the modications, and so
on. Clearly, in this iterative process, statistics plays a major role with its emphasis on
proper design and analysis of experiments and the resulting inferences upon which
decisions can be made. Through statistics, information is provided that is relevant to
taking certain actions, including improving manufactured products, providing better
services, marketing new products or services, forecasting energy needs, classifying
diseases better, and so on.
Statisticians recognize that there are often errors in their inferences, and they
attempt to quantify the probabilities of those mistakes and make them as small as
possible. That these uncertainties even exist is due to the fact that there is variation
in the data. Even though experiments are repeated under seemingly the same condi-
tions, the results vary from trial to trial. In light of this uncertainty, the statistician tries
to summarize the data in the best possible way, always explaining the error structures
of the statistical estimates.
This is an important lesson to be learned: Variation is almost everywhere. It is
the statistician’s job to understand variation. Often, as in manufacturing, the desire is
to reduce variation so that the products will be more consistent. In other words, car
doors will t better in the manufacturing of automobiles if the variation is decreased
by making each door closer to its target values.
ix
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x Prologue
Any student of statistics should understand the nature of variability and the
necessity for creating probabilistic models of that variability. We cannot avoid making
inferences and decisions in the face of this uncertainty; however, these inferences and
decisions are greatly inuenced by the probabilistic models selected. Some persons
are better model builders than others and accordingly will make better inferences and
decisions. The assumptions needed for each statistical model are carefully examined;
it is hoped that thereby the reader will become a better model builder.
Finally, we must mention how modern statistical analyses have become depen-
dent upon the computer. Increasingly, statisticians and computer scientists are
working together in areas of exploratory data analysis and “data mining.” Statistical
software development is critical today, for the best of it is needed in complicated
data analyses. In light of this growing relationship between these two elds, it is good
advice for bright students to take substantial offerings in statistics and in computer
science.
Students majoring in statistics, computer science, or a program at their interface
such as data science are in great demand in the workplace and in graduate pro-
grams. Clearly, they can earn advanced degrees in statistics or computer science or
both. But, more important, they are highly desirable candidates for graduate work
in other areas: actuarial science, industrial engineering, nance, marketing, account-
ing, management science, psychology, economics, law, sociology, medicine, health
sciences, etc. So many elds have been “mathematized” that their programs are beg-
ging for majors in statistics or computer science. Often, such students become “stars”
in these other areas. We truly hope that we can interest students enough that they
want to study more statistics. If they do, they will nd that the opportunities for very
successful careers are numerous.
