Probability E-Book

John J. Kinney

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Key Concepts in Philosophy
• Sprache: Englisch

Seitenzahl: 645

Veröffentlichungsjahr: 2015

Beschreibung

Praise for the First Edition

"This is a well-written and impressively presented introduction to probability and statistics. The text throughout is highly readable, and the author makes liberal use of graphs and diagrams to clarify the theory." - The Statistician

Thoroughly updated, Probability: An Introduction with Statistical Applications, Second Edition features a comprehensive exploration of statistical data analysis as an application of probability. The new edition provides an introduction to statistics with accessible coverage of reliability, acceptance sampling, confidence intervals, hypothesis testing, and simple linear regression. Encouraging readers to develop a deeper intuitive understanding of probability, the author presents illustrative geometrical presentations and arguments without the need for rigorous mathematical proofs.

The Second Edition features interesting and practical examples from a variety of engineering and scientific fields, as well as:

• Over 880 problems at varying degrees of difficulty allowing readers to take on more challenging problems as their skill levels increase
• Chapter-by-chapter projects that aid in the visualization of probability distributions
• New coverage of statistical quality control and quality production
• An appendix dedicated to the use of Mathematica® and a companion website containing the referenced data sets

Featuring a practical and real-world approach, this textbook is ideal for a first course in probability for students majoring in statistics, engineering, business, psychology, operations research, and mathematics. Probability: An Introduction with Statistical Applications, Second Edition is also an excellent reference for researchers and professionals in any discipline who need to make decisions based on data as well as readers interested in learning how to accomplish effective decision making from data.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

Preface for the First Edition

Historical Note

About the Text

For the Instructor

Preface for the Second Edition

Chapter 1: Sample Spaces and Probability

1.1 Discrete Sample Spaces

1.2 Events; Axioms of Probability

1.3 Probability Theorems

1.4 Conditional Probability and Independence

1.5 Some Examples

1.6 Reliability of Systems

1.7 Counting Techniques

1.8 Chapter Review

1.9 PROBLEMS FOR REVIEW

Chapter 2: Discrete Random Variables and Probability Distributions

2.1 Random Variables

2.2 Distribution Functions

2.3 Expected Values of Discrete Random Variables

2.4 Binomial Distribution

2.5 A Recursion

2.6 Some Statistical Considerations

2.7 Hypothesis Testing: Binomial Random Variables

2.8 Distribution of A Sample Proportion

2.9 Geometric and Negative Binomial Distributions

2.10 The Hypergeometric Random Variable: Acceptance Sampling

2.11 Acceptance Sampling (Continued)

2.12 The Hypergeometric Random Variable: Further Examples

2.13 The Poisson Random Variable

2.14 The Poisson Process

Chapter Review

Problems for Review

Chapter 3: Continuous Random Variables and Probability Distributions

3.1 Introduction

3.2 Uniform Distribution

3.3 Exponential Distribution

3.4 Reliability

3.5 Normal Distribution

3.6 Normal Approximation to the Binomial Distribution

3.7 Gamma and Chi-Squared Distributions

3.8 Weibull Distribution

Chapter Review

Problems For Review

Chapter 4: Functions of Random Variables; Generating Functions; Statistical Applications

4.1 Introduction

4.2 Some Examples of Functions of Random Variables

4.3 Probability Distributions of Functions of Random Variables

4.4 Sums of Random Variables I

4.5 Generating Functions

4.6 Some Properties of Generating Functions

4.7 Probability Generating Functions for Some Specific Probability Distributions

4.8 Moment Generating Functions

4.9 Properties of Moment Generating Functions

4.10 Sums of Random Variables—II

4.11 The Central Limit Theorem

4.12 Weak Law of Large Numbers

4.13 Sampling Distribution of the Sample Variance

4.14 Hypothesis Tests and Confidence Intervals for a Single Mean

4.15 Hypothesis Tests on Two Samples

4.16 Least Squares Linear Regression

4.17 Quality Control Chart for

Chapter Review

Problems for Review

Chapter 5: Bivariate Probability Distributions

5.1 Introduction

5.2 Joint and Marginal Distributions

5.3 Conditional Distributions and Densities

5.4 Expected Values and the Correlation Coefficient

5.5 Conditional Expectations

5.6 Bivariate Normal Densities

5.7 Functions of Random Variables

CHAPTER REVIEW

PROBLEMS FOR REVIEW

Chapter 6: Recursions and Markov Chains

6.1 Introduction

6.2 Some Recursions and their Solutions

6.3 Random Walk and Ruin

6.4 Waiting Times for Patterns in Bernoulli Trials

6.5 Markov Chains

CHAPTER REVIEW

PROBLEMS FOR REVIEW

Chapter 7: Some Challenging Problems

7.1 My Socks and

7.2 Expected Value

7.3 Variance

7.4 Other “Socks” Problems

7.5 Coupon Collection and Related Problems

7.6 Conclusion

7.7 Jackknifed Regression and the Bootstrap

7.8 Cook's Distance

7.9 The Bootstrap

7.10 On Waldegrave's Problem

7.11 Probabilities of Winning

7.12 More than Three Players

7.13 Conclusion

7.14 On Huygen's First Problem

7.15 Changing the Sums for the Players

Bibliography: Where to Learn More

Appendix A: Use of Mathematica in Probability and Statistics

Chapter One

Chapter Two

Chapter Three

Chapter Four

Chapter Five

Chapter Six

Appendix B: Answers for Odd-Numbered Exercises

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Appendix C: Standard Normal Distribution

The Distribution Table

Chi-Squared Distribution Table

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface for the First Edition

Chapter 1: Sample Spaces and Probability

List of Illustrations

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 1.6

Figure 1.7

Figure 1.8

Figure 1.9

Figure 1.10

Figure 1.11

Figure 1.12

Figure 1.15

Figure 1.16

Figure 1.17

Figure 1.18

Figure 1.19

Figure 1.20

Figure 1.21

Figure 1.22

Figure 1.23

Figure 1.24

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

Figure 2.15

Figure 2.16

Figure 2.17

Figure 2.18

Figure 2.19

Figure 2.20

Figure 2.21

Figure 2.22

Figure 2.23

Figure 2.24

Figure 2.25

Figure 2.26

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16

Figure 3.17

Figure 3.18

Figure 3.19

Figure 4.1

Figure 4.18

Figure 4.19

Figure 4.20

Figure 4.21

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure 5.16

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 7.1

List of Tables

Table 2.1

Table 5.1

Table 7.1

Table 7.2

Table 7.3

Table 7.4

Probability

An Introduction with Statistical Applications

Second Edition

John J. Kinney

Colorado Springs, CO

Copyright © 2015 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

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Library of Congress Cataloging-in-Publication Data:

Kinney, John J.

Probability : an introduction with statistical applications / John Kinney, Colorado Springs,

CO. – Second edition.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-94708-1 (cloth)

1. Probabilities–Textbooks. 2. Mathematical statistics–Textbooks. I. Title.

QA273.K493 2015

519.2–dc23

2014020218

This book is for

Cherry and Kaylyn

Preface for the First Edition

Historical Note

The theory of probability is concerned with events that occur when randomness or chance influences the result. When the data from a sample survey or the occurrence of extreme weather patterns are common enough examples of situations where randomness is involved, we have come to presume that many models of the physical world contain elements of randomness as well. Scientists now commonly suppose that their models contain random components as well as deterministic components. Randomness, of course, does not involve any new physical forces; rather than measuring all the forces involved and thus predicting the exact outcome of an experiment, we choose to combine all these forces and call the result random. The study of random events is the subject of this book.

It is impossible to chronicle the first interest in events involving randomness or chance, but we do know of a correspondence between Blaise Pascal and Pierre de Fermat in the middle of the seventeenth century regarding questions arising in gambling games. Appropriate mathematical tools for the analysis of such situations were not available at that time, but interest continued among some mathematicians. For a long time, the subject was connected only to gambling games and its development was considerably restricted by the situations arising from such considerations. Mathematical techniques suitable for problems involving randomness have produced a theory applicable to not only gambling situations but also more practical situations. It has not been until recent years, however, that scientists and engineers have become increasingly aware of the presence of random factors in their experiments and manufacturing processes and have become interested in measuring or controlling these factors.

It is the realization that the statistical analysis of experimental data, based on the theory of probability, is of great importance to experimenters that has brought the theory to the forefront of applicable mathematics. The history of probability and the statistical analysis it makes possible illustrate a prime example of seemingly useless mathematical research that now has an incredibly wide range of practical application. Mathematical models for experimental situations now commonly involve both deterministic and random terms. It is perhaps a simplification to say that science, while interested in deterministic models to explain the physical world, now is interested as well in separating deterministic factors from random factors and measuring their relative importance.

There are two facts that strike me as most remarkable about the theory of probability. One is the apparent contradiction that random events are in reality well behaved and that there are laws of probability. The outcome on one toss of a coin cannot be predicted, but given 10,000 tosses of the same coin, many events can be predicted with a high degree of accuracy. The second fact, which the reader will soon perceive, is the pervasiveness of a probability distribution known as the normal distribution. This distribution, which will be defined and discussed at some length, arises in situations which at first glance have little in common: the normal distribution is an essential tool in statistical modeling and is perhaps the single most important concept in statistical inference.

There are reasons for this, and it is my purpose to explain these in this book.

About the Text

From the author's perspective, the characteristics of this text which most clearly differentiate it from others currently available include the following:

Applications to a variety of scientific fields, including engineering, appear in every chapter.

Integration of computer algebra systems such as Mathematica provides insight into both the structure and results of problems in probability.

A great variety of problems at varying levels of difficulty provides a desirable flexibility in assignments.

Topics in statistics appear throughout the text so that professors can include or omit these as the nature of their course warrants.

Some problems are structured and solved using recursions since computers and computer algebra systems facilitate this.

Significant and practical topics in quality control and quality production are introduced.

It has been my purpose to write a book that is readable by students who have some background in multivariable calculus. Mathematical ideas are often easily understood until one sees formal definitions that frequently obscure such understanding. Examples allow us to explore ideas without the burden of language. Therefore, I often begin with examples and follow with the ideas motivated first by them; this is quite purposeful on my part, since language often obstructs understanding of otherwise simply perceived notions.

I have attempted to give examples that are interesting and often practical in order to show the widespread applicability of the subject. I have sometimes sacrificed exact mathematical precision for the sake of readability; readers who seek a more advanced explication of the subject will have no trouble in finding suitable sources. I have proceeded in the belief that beginning students want most to know what the subject encompasses and for what it may be useful. More theoretical courses may then be chosen as time and opportunity allow. For those interested, the bibliography contains a number of current references.

An author has considerable control over the reader by selecting the material, its order of presentation, and the explication. I am hopeful that I have executed these duties with due regard for the reader. While the author may not be described with any sort of precision as the holder of a tightrope, I have been guided by the admonition: “It's not healthy for the tightrope walker to be misunderstood by the person who's holding the rope.”1

The book makes free use of the now widely available computer algebra systems. I have used Mathematica, Maple, and Derive for various problems and examples in the book, and I hope the reader has access to one of these marvelous mathematical aids. These systems allow us the incredible opportunity to see graphs and surfaces easily, which otherwise would be very difficult and time-consuming to produce. Computer algebra systems make some parts of mathematics visual and thereby add immensely to our understanding. Derivatives, integrals, series expansions, numerical computation, and the solution of recursions are used throughout the book, but the reader will find that only the results are included: in my opinion there is no longer any reason to dwell on calculation of either a numeric or algebraic sort. We can now concentrate on the meaning of the results without being restrained by the often mechanical effort in achieving them; hence our concentration is on the structure of the problem and the insight the solution gives. Graphs are freely drawn and, when appropriate, a geometric view of the problem is given so that the solution and the problem can be visualized. Numerical approximations are given when exact solutions are not feasible. The reader without a computer algebra system can still do the problems; the reader with such a system can reproduce every graph in the book exactly as it appears. I have included a fairly expensive appendix in which computer commands in Mathematica are given for many of the examples in which Mathematica was used; this should also ease the translation to other computer algebra systems. The reader with access to a computer algebra system should refer to Appendix 1 fairly frequently.

Although I hope the book is readable and as completely explanatory as a probability text may be, I know that students often do not read the text, but proceed directly to the problems. There is nothing wrong with this; after all, if the ability to solve practical problems is the goal, then the student who can do this without reading the text is to be admired. Readers are warned, however, that probability problems are rarely repetitive; the solution of one problem does not necessarily give even any sort of hint as to the solution of the next problem. I have included over 840 problems so that a reader who solves the problems can be reasonably assured that the concepts involving them are understood.

The problem sections begin with the easiest problems and gradually work their way up to some reasonably difficult problems while remaining within the scope and level of the book. In discussing a forthcoming examination with my students, I summarize the material and give some suggestions for practice problems, so I have followed each chapter by a Chapter Summary, some suggestions for Review Problems, and finally some Supplementary Problems.

For the Instructor

Texts on probability often use generating functions and recursions in the solution of many complex problems; with our use of computer algebra systems, we can determine generating functions, and often their power series expansions, with ease. The structure of generating functions is also used to explain limiting behavior in many situations. Many interesting problems can be best described in terms of recursions; since computer algebra systems allow us to solve such recursions, some discussion of recursive functions is given. Proofs are often given using recursions, a novel feature of the book. Occasionally, the more traditional proofs are given in the exercises.

Although numerous applications of the theory are given in the text and in the problems, the text by no means exhausts the applications of the theory of probability. In addition to solving many practical and varied problems, the theory of probability also provides the basis for the theory of statistical inference and the analysis of data. Statistical analysis is combined with the theory of probability throughout the book. Hypothesis testing, confidence intervals, acceptance sampling, and control charts are considered at various points in the text. The order in which these topics are to be considered is entirely up to the instructor; the book is quite flexible in allowing sections to be skipped, or delayed, resulting in rearrangement of the material. This book will serve as a first introduction to statistics, but the reader who intends to apply statistics should also elect a course in applied statistics. In my opinion, statistics will be the centerpiece of applied mathematics in the twenty-first century.

1

Smilla's Sense of Snow

, by Peter Hoeg (Farrar, Straus and Giroux: New York, 1993).

Preface for the Second Edition

I am pleased to offer a second edition of this text. The reasons for writing the book remain the same and are indicated in the preface for the first edition. While remaining readable and I hope useful for both the student and the instructor, I want to point out some differences between the two editions.

The first edition was written when Mathematica was in its fourth release; it is now in its ninth release and while its capabilities have grown, some of the commands, especially those regarding graphs, have changed. Therefore, Appendix 1 is totally new, reflecting the changes in Mathematica.

Both first and second editions contain about 120 graphs; these have been mostly redrawn.

The problems are of primary importance to the student. Being able to solve them verifies the student's mastery of the material. The book now contains over 880 problems, 60 or so of which are new.

Chapter 7, titled “Some Challenging Problems”, is new. Five problems, or sets of problems, some of which have been studied by famous mathematicians, are introduced. Open questions are given, some of which will challenge the reader. Problems are almost always capable of extension; the reader may do this while doing a project regarding one of the major problems.

I have profited from comments from both instructors and students who used the first edition. In a sense I owe a debt to every student of mine at Rose–Hulman Institute of Technology. Heartfelt Thank yous go to Sari Freedman and my editor, Susanne Steitz-Filler of John Wiley & Sons. Sangeetha Parthasarathy of LaserWords has been very helpful and patient during the production process. I have been fortunate to rely on the extensive computer skills of my nephew, Scott Carter to whom I owe a big Thank You. But I owe the greatest debt to my wife, Cherry, who has out up with my long hours in the study. I also owe a pat on the head for Ginger who allowed me to refresh while guiding me on long walks through our Old North End neighborhood.

JOHN J. KINNEY

March 4, 2014

Colorado Springs

Chapter 1Sample Spaces and Probability

1.1 Discrete Sample Spaces

Probability theory deals with situations in which there is an element of randomness or chance. Some models of the physical world are deterministic, that is, they predict exactly what will happen under certain circumstances. For example, if an object is dropped from a height and given no initial velocity, its distance, , from the starting point is given by , where is the acceleration due to gravity and is the time. If one tried to apply the formula in a practical situation, one would not find very satisfactory results. The problem is that the formula applies only in a vacuum and ignores the shape of the object and the resistance of the air as well as other factors. Although some of these factors can be determined, we generally combine them and say that the result has a random or chance component. Our model then becomes , where denotes the random component of the model. In contrast with the deterministic model, this model is stochastic.

Science often considers stochastic models; in formulating new models, the scientist may try to determine the contributions of both deterministic and random components of the model in predicting accurate results.

The mathematical theory of probability arose in consideration of games of chance, but, as the above-mentioned example shows, it is now widely used in far more practical and applied situations. We encounter other circumstances frequently in everyday life in which we presume that some random factors are at work. Here are some simple examples. What is the chance I will find that all eight traffic lights I pass through on my way to work are green? What are my chances for winning a lottery? I have a ten-volume encyclopedia that I have packed in separate boxes. If the boxes become mixed up and I draw the volumes out at random, what is the chance that my encyclopedia will be in order? My desk lamp has a bulb that is “guaranteed” to last 5000 hours. It has been used for 3000 hours. What is the chance that I must replace it before 2000 more hours are used? Each of these situations involves a random event whose specific outcome is unpredictable in advance.