Aims Community College Random Variables and Distributions Discussion Questions I am having difficult to solve 6 questions from “Random Variables and Distri

Aims Community College Random Variables and Distributions Discussion Questions I am having difficult to solve 6 questions from “Random Variables and Distributions” because of this COVID 19 lockdown online class. If you need a book then I will attach book file as a .pdf. Probability and Statistics
Fourth Edition
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Probability and Statistics
Fourth Edition
Morris H. DeGroot
Carnegie Mellon University
Mark J. Schervish
Carnegie Mellon University
Addison-Wesley
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Library of Congress Cataloging-in-Publication Data
DeGroot, Morris H., 1931–1989.
Probability and statistics / Morris H. DeGroot, Mark J. Schervish.—4th ed.
p. cm.
ISBN 978-0-321-50046-5
1. Probabilities—Textbooks. 2. Mathematical statistics—Textbooks.
I. Schervish, Mark J. II. Title.
QA273.D35 2012
519.2—dc22
2010001486
Copyright © 2012, 2002 Pearson Education, 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
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please submit a written request to Pearson Education, Inc., Rights and Contracts Department,
75 Arlington Street, Suite 300, Boston, MA 02116, fax your request to 617-848-7047, or e-mail
at http://www.pearsoned.com/legal/permissions.htm.
1 2 3 4 5 6 7 8 9 10—EB—14 13 12 11 10
ISBN 10: 0-321-50046-6
www.pearsonhighered.com
ISBN 13: 978-0-321-50046-5
To the memory of Morrie DeGroot.
MJS
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Contents
Preface
1
xi
Introduction to Probability
1.1
The History of Probability
1.2
Interpretations of Probability
1.3
Experiments and Events
1.4
Set Theory
1.5
The Definition of Probability
1.6
Finite Sample Spaces
1.7
Counting Methods
1.8
Combinatorial Methods
1.9
Multinomial Coefficients
1
1
2
5
6
16
22
25
32
42
1.10 The Probability of a Union of Events
1.11 Statistical Swindles
51
1.12 Supplementary Exercises
2
53
Conditional Probability

55
2.1
The Definition of Conditional Probability
2.2
Independent Events
2.3
Bayes’ Theorem
2.4
2.5
3
46
55
66
76
The Gambler’s Ruin Problem
Supplementary Exercises
86
90
Random Variables and Distributions
3.1
Random Variables and Discrete Distributions
3.2
Continuous Distributions
3.3
The Cumulative Distribution Function
3.4
Bivariate Distributions
118
3.5
Marginal Distributions
130
3.6
Conditional Distributions
141
3.7
Multivariate Distributions
152
3.8
Functions of a Random Variable
3.9
Functions of Two or More Random Variables
3.10 Markov Chains
100
188
3.11 Supplementary Exercises
vii
93
202
107
167
175
93
viii
Contents
4
Expectation
4.1
The Expectation of a Random Variable
4.2
Properties of Expectations
4.3
Variance
4.4
Moments
4.5
The Mean and the Median
241
4.6
Covariance and Correlation
248
4.7
Conditional Expectation
4.8
4.9
5
207
Utility
217
225
234
256
265
Supplementary Exercises
272
Special Distributions
275
5.1
Introduction
275
5.2
The Bernoulli and Binomial Distributions
5.3
The Hypergeometric Distributions
5.4
The Poisson Distributions
5.5
The Negative Binomial Distributions
5.6
The Normal Distributions
302
5.7
The Gamma Distributions
316
5.8
The Beta Distributions
5.9
The Multinomial Distributions
287
297
327
5.11 Supplementary Exercises
7
333
337
345
Large Random Samples
347
6.1
Introduction
6.2
The Law of Large Numbers
348
6.3
The Central Limit Theorem
360
6.4
The Correction for Continuity
6.5
Supplementary Exercises
Estimation
275
281
5.10 The Bivariate Normal Distributions
6
207
347
371
375
376
7.1
Statistical Inference
376
7.2
Prior and Posterior Distributions
7.3
Conjugate Prior Distributions
7.4
Bayes Estimators
408
385
394
Contents
7.5
Maximum Likelihood Estimators
7.6
Properties of Maximum Likelihood Estimators
7.7
Sufficient Statistics
7.8
Jointly Sufficient Statistics
7.9
Improving an Estimator
449
455
461
Sampling Distributions of Estimators
The Sampling Distribution of a Statistic
8.2
The Chi-Square Distributions
8.3
Joint Distribution of the Sample Mean and Sample Variance
8.4
The t Distributions
8.5
Confidence Intervals
8.7
8.8
8.9
464
469
485
Bayesian Analysis of Samples from a Normal Distribution
Unbiased Estimators
Fisher Information
473
480
495
506
514
Supplementary Exercises
528
Testing Hypotheses
9.1
530
Problems of Testing Hypotheses
9.2
Testing Simple Hypotheses
9.3
Uniformly Most Powerful Tests
9.4
Two-Sided Alternatives
530
550
559
567
9.5
The t Test
576
9.6
Comparing the Means of Two Normal Distributions
9.7
The F Distributions
9.8
Bayes Test Procedures
9.9
Foundational Issues
587
597
605
617
9.10 Supplementary Exercises
10
464
8.1
8.6
9
426
443
7.10 Supplementary Exercises
8
417
621
Categorical Data and Nonparametric Methods
10.1 Tests of Goodness-of-Fit
624
10.2 Goodness-of-Fit for Composite Hypotheses
10.3 Contingency Tables
641
10.4 Tests of Homogeneity
10.5 Simpson’s Paradox
647
653
10.6 Kolmogorov-Smirnov Tests
657
633
624
ix
x
Contents
10.7 Robust Estimation
10.8 Sign and Rank Tests
666
678
10.9 Supplementary Exercises
11
686
Linear Statistical Models
11.1 The Method of Least Squares
11.2 Regression
689
689
698
11.3 Statistical Inference in Simple Linear Regression
11.4 Bayesian Inference in Simple Linear Regression
11.5 The General Linear Model and Multiple Regression
11.6 Analysis of Variance
754
11.7 The Two-Way Layout
763
11.8 The Two-Way Layout with Replications
11.9 Supplementary Exercises
12
Simulation
783
787
12.1 What Is Simulation?
787
12.2 Why Is Simulation Useful?
791
12.3 Simulating Specific Distributions
804
12.4 Importance Sampling
816
12.5 Markov Chain Monte Carlo
823
12.6 The Bootstrap
839
12.7 Supplementary Exercises
Tables
850
853
Answers to Odd-Numbered Exercises
References
Index
879
885
865
772
707
729
736
Preface
Changes to the Fourth Edition
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.
I have reorganized many main results that were included in the body of the
text by labeling them as theorems in order to facilitate students in finding and
referencing these results.
I have pulled the important defintions and assumptions out of the body of the
text and labeled them as such so that they stand out better.
When a new topic is introduced, I introduce it with a motivating example before
delving into the mathematical formalities. Then I return to the example to
illustrate the newly introduced material.
I moved the material on the law of large numbers and the central limit theorem
to a new Chapter 6. It seemed more natural to deal with the main large-sample
results together.
I moved the section on Markov chains into Chapter 3. Every time I cover this
material with my own students, I stumble over not being able to refer to random
variables, distributions, and conditional distributions. I have actually postponed
this material until after introducing distributions, and then gone back to cover
Markov chains. I feel that the time has come to place it in a more natural
location. I also added some material on stationary distributions of Markov
chains.
I have moved the lengthy proofs of several theorems to the ends of their
respective sections in order to improve the flow of the presentation of ideas.
I rewrote Section 7.1 to make the introduction to inference clearer.
I rewrote Section 9.1 as a more complete introduction to hypothesis testing,
including likelihood ratio tests. For instructors not interested in the more mathematical theory of hypothesis testing, it should now be easier to skip from
Section 9.1 directly to Section 9.5.
Some other changes that readers will notice:
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.
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.
.
.
I have replaced the notation in which the intersection of two sets A and B had
been represented AB with the more popular A ∩ B. The old notation, although
mathematically sound, seemed a bit arcane for a text at this level.
I added the statements of Stirling’s formula and Jensen’s inequality.
I moved the law of total probability and the discussion of partitions of a sample
space from Section 2.3 to Section 2.1.
I define the cumulative distribution function (c.d.f.) as the prefered name of
what used to be called only the distribution function (d.f.).
I added some discussion of histograms in Chapters 3 and 6.
I rearranged the topics in Sections 3.8 and 3.9 so that simple functions of random
variables appear first and the general formulations appear at the end to make
it easier for instructors who want to avoid some of the more mathematically
challenging parts.
I emphasized the closeness of a hypergeometric distribution with a large number of available items to a binomial distribution.
xi
xii
Preface
.
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.
.
.
.
I gave a brief introduction to Chernoff bounds. These are becoming increasingly
important in computer science, and their derivation requires only material that
is already in the text.
I changed the definition of confidence interval to refer to the random interval
rather than the observed interval. This makes statements less cumbersome, and
it corresponds to more modern usage.
I added a brief discussion of the method of moments in Section 7.6.
I added brief introductions to Newton’s method and the EM algorithm in
Chapter 7.
I introduced the concept of pivotal quantity to facilitate construction of confidence intervals in general.
I added the statement of the large-sample distribution of the likelihood ratio
test statistic. I then used this as an alternative way to test the null hypothesis
that two normal means are equal when it is not assumed that the variances are
equal.
I moved the Bonferroni inequality into the main text (Chapter 1) and later
(Chapter 11) used it as a way to construct simultaneous tests and confidence
intervals.
How to Use This Book
The text is somewhat long for complete coverage in a one-year course at the undergraduate level and is designed so that instructors can make choices about which topics
are most important to cover and which can be left for more in-depth study. As an example, many instructors wish to deemphasize the classical counting arguments that
are detailed in Sections 1.7–1.9. An instructor who only wants enough information
to be able to cover the binomial and/or multinomial distributions can safely discuss only the definitions and theorems on permutations, combinations, and possibly
multinomial coefficients. Just make sure that the students realize what these values
count, otherwise the associated distributions will make no sense. The various examples in these sections are helpful, but not necessary, for understanding the important
distributions. Another example is Section 3.9 on functions of two or more random
variables. The use of Jacobians for general multivariate transformations might be
more mathematics than the instructors of some undergraduate courses are willing
to cover. The entire section could be skipped without causing problems later in the
course, but some of the more straightforward cases early in the section (such as convolution) might be worth introducing. The material in Sections 9.2–9.4 on optimal
tests in one-parameter families is pretty mathematics, but it is of interest primarily
to graduate students who require a very deep understanding of hypothesis testing
theory. The rest of Chapter 9 covers everything that an undergraduate course really
needs.
In addition to the text, the publisher has an Instructor’s Solutions Manual, available for download from the Instructor Resource Center at www.pearsonhighered
.com/irc, which includes some specific advice about many of the sections of the text.
I have taught a year-long probability and statistics sequence from earlier editions of
this text for a group of mathematically well-trained juniors and seniors. In the first
semester, I covered what was in the earlier edition but is now in the first five chapters (including the material on Markov chains) and parts of Chapter 6. In the second
semester, I covered the rest of the new Chapter 6, Chapters 7–9, Sections 11.1–11.5,
and Chapter 12. I have also taught a one-semester probability and random processes
Preface
xiii
course for engineers and computer scientists. I covered what was in the old edition
and is now in Chapters 1–6 and 12, including Markov chains, but not Jacobians. This
latter course did not emphasize mathematical derivation to the same extent as the
course for mathematics students.
A number of sections are designated with an asterisk (*). This indicates that
later sections do not rely materially on the material in that section. This designation
is not intended to suggest that instructors skip these sections. Skipping one of these
sections will not cause the students to miss definitions or results that they will need
later. The sections are 2.4, 3.10, 4.8, 7.7, 7.8, 7.9, 8.6, 8.8, 9.2, 9.3, 9.4, 9.8, 9.9, 10.6,
10.7, 10.8, 11.4, 11.7, 11.8, and 12.5. Aside from cross-references between sections
within this list, occasional material from elsewhere in the text does refer back to
some of the sections in this list. Each of the dependencies is quite minor, however.
Most of the dependencies involve references from Chapter 12 back to one of the
optional sections. The reason for this is that the optional sections address some of
the more difficult material, and simulation is most useful for solving those difficult
problems that cannot be solved analytically. Except for passing references that help
put material into context, the dependencies are as follows:
.
.
.
The sample distribution function (Section 10.6) is reintroduced during the
discussion of the bootstrap in Section 12.6. The sample distribution function
is also a useful tool for displaying simulation results. It could be introduced as
early as Example 12.3.7 simply by covering the first subsection of Section 10.6.
The material on robust estimation (Section 10.7) is revisited in some simulation
exercises in Section 12.2 (Exercises 4, 5, 7, and 8).
Example 12.3.4 makes reference to the material on two-way analysis of variance
(Sections 11.7 and 11.8).
Supplements
The text is accompanied by the following supplementary material:
.
.
Instructor’s Solutions Manual contains fully worked solutions to all exercises
in the text. Available for download from the Instructor Resource Center at
www.pearsonhighered.com/irc.
Student Solutions Manual contains fully worked solutions to all odd exercises in
the text. Available for purchase from MyPearsonStore at www.mypearsonstore
.com. (ISBN-13: 978-0-321-71598-2; ISBN-10: 0-321-71598-5)
Acknowledgments
There are many people that I want to thank for their help and encouragement during
this revision. First and foremost, I want to thank Marilyn DeGroot and Morrie’s
children for giving me the chance to revise Morrie’s masterpiece.
I am indebted to the many readers, reviewers, colleagues, staff, and people
at Addison-Wesley whose help and comments have strengthened this edition. The
reviewers were:
Andre Adler, Illinois Institute of Technology; E. N. Barron, Loyola University; Brian
Blank, Washington University in St. Louis; Indranil Chakraborty, University of Oklahoma; Daniel Chambers, Boston College; Rita Chattopadhyay, Eastern Michigan
University; Stephen A. Chiappari, Santa Clara University; Sheng-Kai Chang, Wayne
State University; Justin Corvino, Lafayette College; Michael Evans, University of
xiv
Preface
Toronto; Doug Frank, Indiana University of Pennsylvania; Anda Gadidov, Kennesaw State University; Lyn Geisler, Randolph–Macon College; Prem Goel, Ohio
State University; Susan Herring, Sonoma State University; Pawel Hitczenko, Drexel
University; Lifang Hsu, Le Moyne College; Wei-Min Huang, Lehigh University;
Syed Kirmani, University of Northern Iowa; Michael Lavine, Duke University; Rich
Levine, San Diego State University; John Liukkonen, Tulane University; Sergio
Loch, Grand View College; Rosa Matzkin, Northwestern University; Terry McConnell, Syracuse University; Hans-Georg Mueller, University of California–Davis;
Robert Myers, Bethel College; Mario Peruggia, The Ohio State University; Stefan
Ralescu, Queens University; Krishnamurthi Ravishankar, SUNY New Paltz; Diane
Saphire, Trinity University; Steven Sepanski, Saginaw Valley State University; HenSiong Tan, Pennsylvania University; Kanapathi Thiru, University of Alaska; Kenneth Troske, Johns Hopkins University; John Van Ness, University of Texas at Dallas; Yehuda Vardi, Rutgers University; Yelena Vaynberg, Wayne State University;
Joseph Verducci, Ohio State University; Mahbobeh Vezveai, Kent State University;
Brani Vidakovic, Duke University; Karin Vorwerk, Westfield State College; Bette
Warren, Eastern Michigan University; Calvin L. Williams, Clemson University; Lori
Wolff, University of Mississippi.
The person who checked the accuracy of the book was Anda Gadidov, Kennesaw State University. I would also like to thank my colleagues at Carnegie Mellon
University, especially Anthony Brockwell, Joel Greenhouse, John Lehoczky, Heidi
Sestrich, and Valerie Ventura.
The people at Addison-Wesley and other organizations that helped produce
the book were Paul Anagnostopoulos, Patty Bergin, Dana Jones Bettez, Chris
Cummings, Kathleen DeChavez, Alex Gay, Leah Goldberg, Karen Hartpence, and
Christina Lepre.
If I left anyone out, it was unintentional, and I apologize. Errors inevitably arise
in any project like this (meaning a project in which I am involved). For this reason,
I shall post information about the book, including a list of corrections, on my Web
page, http://www.stat.cmu.edu/~mark/, as soon as the book is published. Readers are
encouraged to send me any errors that they discover.
Mark J. Schervish
October 2010
Chapter
Introduction to
Probability
1.1
1.2
1.3
1.4
1.5
1.6
1
The History of Probability
Interpretations of Probability
Experiments and Events
Set Theory
The Definition of Probability
Finite Sample Spaces
1.7
1.8
1.9
1.10
1.11
1.12
Counting Methods
Combinatorial Methods
Multinomial Coefficients
The Probability of a Union of Events
Statistical Swindles
Supplementary Exercises
1.1 The History of Probability
The use of probability to measure uncertainty and variability dates back hundreds
of years. Probability has found application in areas as diverse as medicine, gambling, weather forecasting, and the law.
The concepts of chance and uncertainty are as old as civilization itself. People have
alway…
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