Springer Texts in Statistics
Jay L. Devore
Kenneth N. Berk
Matthew A. Carlton
Modern
Mathematical
Statistics
with Applications
Third Edition
,Springer Texts in Statistics
Series Editors
G. Allen, Department of Statistics, Houston, TX, USA
R. De Veaux, Department of Mathematics and Statistics, Williams College,
Williamstown, MA, USA
R. Nugent, Department of Statistics, Carnegie Mellon University, Pittsburgh,
PA, USA
,Springer Texts in Statistics (STS) includes advanced textbooks from 3rd- to
4th-year undergraduate courses to 1st- to 2nd-year graduate courses. Exercise
sets should be included. The series editors are currently Genevera I. Allen,
Richard D. De Veaux, and Rebecca Nugent. Stephen Fienberg, George
Casella, and Ingram Olkin were editors of the series for many years.
More information about this series at http://www.springer.com/series/417
,Jay L. Devore Kenneth N. Berk
• •
Matthew A. Carlton
Modern Mathematical
Statistics
with Applications
Third Edition
123
,Jay L. Devore Kenneth N. Berk
Department of Statistics (Emeritus) Department of Mathematics (Emeritus)
California Polytechnic State University Illinois State University
San Luis Obispo, CA, USA Normal, IL, USA
Matthew A. Carlton
Department of Statistics
California Polytechnic State University
San Luis Obispo, CA, USA
ISSN 1431-875X ISSN 2197-4136 (electronic)
Springer Texts in Statistics
ISBN 978-3-030-55155-1 ISBN 978-3-030-55156-8 (eBook)
https://doi.org/10.1007/978-3-030-55156-8
1st edition: © Thomson Brooks Cole, 2007
2nd edition: © Springer Science+Business Media, LLC 2012, corrected publication 2018
3rd edition © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher,
whether the whole or part of the material is concerned, specifically the rights of translation,
reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any
other physical way, and transmission or information storage and retrieval, electronic adaptation,
computer software, or by similar or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are
exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in
this book are believed to be true and accurate at the date of publication. Neither the publisher nor
the authors or the editors give a warranty, expressed or implied, with respect to the material
contained herein or for any errors or omissions that may have been made. The publisher remains
neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
,Preface
Purpose
Our objective is to provide a postcalculus introduction to the discipline of
statistics that
• Has mathematical integrity and contains some underlying theory.
• Shows students a broad range of applications involving real data.
• Is up to date in its selection of topics.
• Illustrates the importance of statistical software.
• Is accessible to a wide audience, including mathematics and statistics
majors (yes, there are quite a few of the latter these days, thanks to the
proliferation of “big data”), prospective engineers and scientists, and
those business and social science majors interested in the quantitative
aspects of their disciplines.
A number of currently available mathematical statistics texts are heavily
oriented toward a rigorous mathematical development of probability and
statistics, with much emphasis on theorems, proofs, and derivations. The
focus is more on mathematics than on statistical practice. Even when applied
material is included, the scenarios are often contrived (many examples and
exercises involving dice, coins, cards, widgets, or a comparison of treatment
A to treatment B).
Our exposition is an attempt to provide a reasonable balance between
mathematical rigor and statistical practice. We believe that showing students
the applicability of statistics to real-world problems is extremely effective in
inspiring them to pursue further coursework and even career opportunities in
statistics. Opportunities for exposure to mathematical foundations will follow
in due course. In our view, it is more important for students coming out of
this course to be able to carry out and interpret the results of a two-sample
t test or simple regression analysis, and appreciate how these are based on
underlying theory, than to manipulate joint moment generating functions or
discourse on various modes of convergence.
v
,vi Preface
Content and Mathematical Level
The book certainly does include core material in probability (Chap. 2),
random variables and their distributions (Chaps. 3–5), and sampling theory
(Chap. 6). But our desire to balance theory with application/data analysis is
reflected in the way the book starts out, with a chapter on descriptive and
exploratory statistical techniques rather than an immediate foray into the
axioms of probability and their consequences. After the distributional
infrastructure is in place, the remaining statistical chapters cover the basics of
inference. In addition to introducing core ideas from estimation and
hypothesis testing (Chaps. 7–10), there is emphasis on checking assumptions
and examining the data prior to formal analysis. Modern topics such as
bootstrapping, permutation tests, residual analysis, and logistic regression are
included. Our treatment of regression, analysis of variance, and categorical
data analysis (Chaps. 11–13) is definitely more oriented to dealing with real
data than with theoretical properties of models. We also show many exam-
ples of output from commonly used statistical software packages, something
noticeably absent in most other books pitched at this audience and level.
The challenge for students at this level should lie with mastery of statis-
tical concepts as well as with mathematical wizardry. Consequently, the
mathematical prerequisites and demands are reasonably modest. Mathemat-
ical sophistication and quantitative reasoning ability are certainly important,
especially as they facilitate dealing with symbolic notation and manipulation.
Students with a solid grounding in univariate calculus and some exposure to
multivariate calculus should feel comfortable with what we are asking
of them. The few sections where matrix algebra appears (transformations in
Chap. 5 and the matrix approach to regression in the last section of Chap. 12)
can easily be deemphasized or skipped entirely. Proofs and derivations are
included where appropriate, but we think it likely that obtaining a conceptual
understanding of the statistical enterprise will be the major challenge for
readers.
Recommended Coverage
There should be more than enough material in our book for a year-long
course. Those wanting to emphasize some of the more theoretical aspects
of the subject (e.g., moment generating functions, conditional expectation,
transformations, order statistics, sufficiency) should plan to spend corre-
spondingly less time on inferential methodology in the latter part of the book.
We have opted not to mark certain sections as optional, preferring instead to
rely on the experience and tastes of individual instructors in deciding what
should be presented. We would also like to think that students could be asked
to read an occasional subsection or even section on their own and then work
exercises to demonstrate understanding, so that not everything would need to
be presented in class. Remember that there is never enough time in a course
of any duration to teach students all that we’d like them to know!
,Preface vii
Revisions for This Edition
• Many of the examples have been updated and/or replaced, especially
those containing real data or references to applications published in
various journals. The same is true of the roughly 1300 exercises in the
book.
• The exposition has been refined and polished throughout to improve
accessibility and eliminate unnecessary material and verbiage. For
example, the categorical data chapter (Chap. 13) has been streamlined by
discarding some of the methodology involving tests when parameters
must be estimated.
• A section on simulation has been added to each of the chapters on
probability, discrete distributions, and continuous distributions.
• The material in the chapter on joint distributions (Chap. 5) has been
reorganized. There is now a separate section on linear combinations and
their properties, and also one on the bivariate normal distribution.
• The material in the chapter on statistics and their sampling distributions
(Chap. 6) has also been reorganized. In particular, there is now a separate
section on the chi-squared, t, and F distributions prior to the one con-
taining derivations of sampling distributions of statistics based on a
normal random sample.
• The chapters on one-sample confidence intervals (Chap. 8) and hypoth-
esis tests (Chap. 9) place more emphasis on t procedures and less on
large-sample z procedures. This is also true of inferences based on two
samples in Chap. 10.
• Chap. 9 now contains a subsection on using the bootstrap to test
hypotheses.
• The material on multiple regression models containing quadratic, inter-
action, and indicator variables has been separated into its own section.
And there is now a separate expanded section on logistic regression.
• The nonparametric and Bayesian material that previously comprised a
single chapter has been separated into two chapters, and material has been
added to each. For example, there is now a section on nonparametric
inferences about population quantiles.
Acknowledgements
We gratefully acknowledge the plentiful feedback provided by reviewers and
colleagues. A special salute goes to Bruce Trumbo for going way beyond his
mandate in providing us an incredibly thoughtful review of 40+ pages
containing many wonderful ideas and pertinent criticisms. Our emphasis on
real data would not have come to fruition without help from the many
individuals who provided us with data in published sources or in personal
communications. We appreciate the production services provided by the
folks at Springer.
,viii Preface
A Final Thought
It is our hope that students completing a course taught from this book will
feel as passionate about the subject of statistics as we still do after so many
years in the profession. Only teachers can really appreciate how gratifying it
is to hear from a student after he or she has completed a course that the
experience had a positive impact and maybe even affected a career choice.
Los Osos, CA, USA Jay L. Devore
Normal, IL, USA Kenneth N. Berk
San Luis Obispo, CA, USA Matthew A. Carlton
, Contents
1 Overview and Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Language of Statistics . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Graphical Methods in Descriptive Statistics . . . . . . . . . . 9
1.3 Measures of Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Measures of Variability . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. 49
2.1 Sample Spaces and Events . . . . . . . . . . ............. 49
2.2 Axioms, Interpretations, and Properties
of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3 Counting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.4 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.6 Simulation of Random Events . . . . . . . . . . . . . . . . . . . . . 94
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3 Discrete Random Variables and Probability
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 111
3.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . .... 111
3.2 Probability Distributions for Discrete Random
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.3 Expected Values of Discrete Random Variables . . . . . . . 126
3.4 Moments and Moment Generating Functions . . . . . . . . . 137
3.5 The Binomial Probability Distribution . . . . . . . . . . . . . . . 144
3.6 The Poisson Probability Distribution . . . . . . . . . . . . . . . . 156
3.7 Other Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . 164
3.8 Simulation of Discrete Random Variables . . . . . . . . . . . . 173
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4 Continuous Random Variables and Probability
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.1 Probability Density Functions and Cumulative
Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.2 Expected Values and Moment Generating Functions . . . 203
4.3 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.4 The Gamma Distribution and Its Relatives . . . . . . . . . . . 230
4.5 Other Continuous Distributions . . . . . . . . . . . . . . . . . . . . 239
ix
Jay L. Devore
Kenneth N. Berk
Matthew A. Carlton
Modern
Mathematical
Statistics
with Applications
Third Edition
,Springer Texts in Statistics
Series Editors
G. Allen, Department of Statistics, Houston, TX, USA
R. De Veaux, Department of Mathematics and Statistics, Williams College,
Williamstown, MA, USA
R. Nugent, Department of Statistics, Carnegie Mellon University, Pittsburgh,
PA, USA
,Springer Texts in Statistics (STS) includes advanced textbooks from 3rd- to
4th-year undergraduate courses to 1st- to 2nd-year graduate courses. Exercise
sets should be included. The series editors are currently Genevera I. Allen,
Richard D. De Veaux, and Rebecca Nugent. Stephen Fienberg, George
Casella, and Ingram Olkin were editors of the series for many years.
More information about this series at http://www.springer.com/series/417
,Jay L. Devore Kenneth N. Berk
• •
Matthew A. Carlton
Modern Mathematical
Statistics
with Applications
Third Edition
123
,Jay L. Devore Kenneth N. Berk
Department of Statistics (Emeritus) Department of Mathematics (Emeritus)
California Polytechnic State University Illinois State University
San Luis Obispo, CA, USA Normal, IL, USA
Matthew A. Carlton
Department of Statistics
California Polytechnic State University
San Luis Obispo, CA, USA
ISSN 1431-875X ISSN 2197-4136 (electronic)
Springer Texts in Statistics
ISBN 978-3-030-55155-1 ISBN 978-3-030-55156-8 (eBook)
https://doi.org/10.1007/978-3-030-55156-8
1st edition: © Thomson Brooks Cole, 2007
2nd edition: © Springer Science+Business Media, LLC 2012, corrected publication 2018
3rd edition © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher,
whether the whole or part of the material is concerned, specifically the rights of translation,
reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any
other physical way, and transmission or information storage and retrieval, electronic adaptation,
computer software, or by similar or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are
exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in
this book are believed to be true and accurate at the date of publication. Neither the publisher nor
the authors or the editors give a warranty, expressed or implied, with respect to the material
contained herein or for any errors or omissions that may have been made. The publisher remains
neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
,Preface
Purpose
Our objective is to provide a postcalculus introduction to the discipline of
statistics that
• Has mathematical integrity and contains some underlying theory.
• Shows students a broad range of applications involving real data.
• Is up to date in its selection of topics.
• Illustrates the importance of statistical software.
• Is accessible to a wide audience, including mathematics and statistics
majors (yes, there are quite a few of the latter these days, thanks to the
proliferation of “big data”), prospective engineers and scientists, and
those business and social science majors interested in the quantitative
aspects of their disciplines.
A number of currently available mathematical statistics texts are heavily
oriented toward a rigorous mathematical development of probability and
statistics, with much emphasis on theorems, proofs, and derivations. The
focus is more on mathematics than on statistical practice. Even when applied
material is included, the scenarios are often contrived (many examples and
exercises involving dice, coins, cards, widgets, or a comparison of treatment
A to treatment B).
Our exposition is an attempt to provide a reasonable balance between
mathematical rigor and statistical practice. We believe that showing students
the applicability of statistics to real-world problems is extremely effective in
inspiring them to pursue further coursework and even career opportunities in
statistics. Opportunities for exposure to mathematical foundations will follow
in due course. In our view, it is more important for students coming out of
this course to be able to carry out and interpret the results of a two-sample
t test or simple regression analysis, and appreciate how these are based on
underlying theory, than to manipulate joint moment generating functions or
discourse on various modes of convergence.
v
,vi Preface
Content and Mathematical Level
The book certainly does include core material in probability (Chap. 2),
random variables and their distributions (Chaps. 3–5), and sampling theory
(Chap. 6). But our desire to balance theory with application/data analysis is
reflected in the way the book starts out, with a chapter on descriptive and
exploratory statistical techniques rather than an immediate foray into the
axioms of probability and their consequences. After the distributional
infrastructure is in place, the remaining statistical chapters cover the basics of
inference. In addition to introducing core ideas from estimation and
hypothesis testing (Chaps. 7–10), there is emphasis on checking assumptions
and examining the data prior to formal analysis. Modern topics such as
bootstrapping, permutation tests, residual analysis, and logistic regression are
included. Our treatment of regression, analysis of variance, and categorical
data analysis (Chaps. 11–13) is definitely more oriented to dealing with real
data than with theoretical properties of models. We also show many exam-
ples of output from commonly used statistical software packages, something
noticeably absent in most other books pitched at this audience and level.
The challenge for students at this level should lie with mastery of statis-
tical concepts as well as with mathematical wizardry. Consequently, the
mathematical prerequisites and demands are reasonably modest. Mathemat-
ical sophistication and quantitative reasoning ability are certainly important,
especially as they facilitate dealing with symbolic notation and manipulation.
Students with a solid grounding in univariate calculus and some exposure to
multivariate calculus should feel comfortable with what we are asking
of them. The few sections where matrix algebra appears (transformations in
Chap. 5 and the matrix approach to regression in the last section of Chap. 12)
can easily be deemphasized or skipped entirely. Proofs and derivations are
included where appropriate, but we think it likely that obtaining a conceptual
understanding of the statistical enterprise will be the major challenge for
readers.
Recommended Coverage
There should be more than enough material in our book for a year-long
course. Those wanting to emphasize some of the more theoretical aspects
of the subject (e.g., moment generating functions, conditional expectation,
transformations, order statistics, sufficiency) should plan to spend corre-
spondingly less time on inferential methodology in the latter part of the book.
We have opted not to mark certain sections as optional, preferring instead to
rely on the experience and tastes of individual instructors in deciding what
should be presented. We would also like to think that students could be asked
to read an occasional subsection or even section on their own and then work
exercises to demonstrate understanding, so that not everything would need to
be presented in class. Remember that there is never enough time in a course
of any duration to teach students all that we’d like them to know!
,Preface vii
Revisions for This Edition
• Many of the examples have been updated and/or replaced, especially
those containing real data or references to applications published in
various journals. The same is true of the roughly 1300 exercises in the
book.
• The exposition has been refined and polished throughout to improve
accessibility and eliminate unnecessary material and verbiage. For
example, the categorical data chapter (Chap. 13) has been streamlined by
discarding some of the methodology involving tests when parameters
must be estimated.
• A section on simulation has been added to each of the chapters on
probability, discrete distributions, and continuous distributions.
• The material in the chapter on joint distributions (Chap. 5) has been
reorganized. There is now a separate section on linear combinations and
their properties, and also one on the bivariate normal distribution.
• The material in the chapter on statistics and their sampling distributions
(Chap. 6) has also been reorganized. In particular, there is now a separate
section on the chi-squared, t, and F distributions prior to the one con-
taining derivations of sampling distributions of statistics based on a
normal random sample.
• The chapters on one-sample confidence intervals (Chap. 8) and hypoth-
esis tests (Chap. 9) place more emphasis on t procedures and less on
large-sample z procedures. This is also true of inferences based on two
samples in Chap. 10.
• Chap. 9 now contains a subsection on using the bootstrap to test
hypotheses.
• The material on multiple regression models containing quadratic, inter-
action, and indicator variables has been separated into its own section.
And there is now a separate expanded section on logistic regression.
• The nonparametric and Bayesian material that previously comprised a
single chapter has been separated into two chapters, and material has been
added to each. For example, there is now a section on nonparametric
inferences about population quantiles.
Acknowledgements
We gratefully acknowledge the plentiful feedback provided by reviewers and
colleagues. A special salute goes to Bruce Trumbo for going way beyond his
mandate in providing us an incredibly thoughtful review of 40+ pages
containing many wonderful ideas and pertinent criticisms. Our emphasis on
real data would not have come to fruition without help from the many
individuals who provided us with data in published sources or in personal
communications. We appreciate the production services provided by the
folks at Springer.
,viii Preface
A Final Thought
It is our hope that students completing a course taught from this book will
feel as passionate about the subject of statistics as we still do after so many
years in the profession. Only teachers can really appreciate how gratifying it
is to hear from a student after he or she has completed a course that the
experience had a positive impact and maybe even affected a career choice.
Los Osos, CA, USA Jay L. Devore
Normal, IL, USA Kenneth N. Berk
San Luis Obispo, CA, USA Matthew A. Carlton
, Contents
1 Overview and Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Language of Statistics . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Graphical Methods in Descriptive Statistics . . . . . . . . . . 9
1.3 Measures of Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Measures of Variability . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. 49
2.1 Sample Spaces and Events . . . . . . . . . . ............. 49
2.2 Axioms, Interpretations, and Properties
of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3 Counting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.4 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.6 Simulation of Random Events . . . . . . . . . . . . . . . . . . . . . 94
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3 Discrete Random Variables and Probability
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 111
3.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . .... 111
3.2 Probability Distributions for Discrete Random
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.3 Expected Values of Discrete Random Variables . . . . . . . 126
3.4 Moments and Moment Generating Functions . . . . . . . . . 137
3.5 The Binomial Probability Distribution . . . . . . . . . . . . . . . 144
3.6 The Poisson Probability Distribution . . . . . . . . . . . . . . . . 156
3.7 Other Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . 164
3.8 Simulation of Discrete Random Variables . . . . . . . . . . . . 173
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4 Continuous Random Variables and Probability
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.1 Probability Density Functions and Cumulative
Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.2 Expected Values and Moment Generating Functions . . . 203
4.3 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.4 The Gamma Distribution and Its Relatives . . . . . . . . . . . 230
4.5 Other Continuous Distributions . . . . . . . . . . . . . . . . . . . . 239
ix
