A First Course in Probability / Edition 6

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More About This Textbook

Overview

This market-leading introduction to probability features exceptionally clear explanations of the mathematics of probability theory and explores its many diverse applications through numerous interesting and motivational examples. The outstanding problem sets are a hallmark feature of this book. Provides clear, complete explanations to fully explain mathematical concepts. Features subsections on the probabilistic method and the maximum-minimums identity. Includes many new examples relating to DNA matching, utility, finance, and applications of the probabilistic method. Features an intuitive treatment of probability—intuitive explanations follow many examples. The Probability Models Disk included with each copy of the book, contains six probability models that are referenced in the book and allow readers to quickly and easily perform calculations and simulations.
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Editorial Reviews

Booknews
A book/disk introduction to probability for students in mathematics, engineering, and the sciences (including the social sciences and management science) who understand elementary calculus. Presents the mathematics of probability theory as well as many examples of applications, covering combinatorial analysis, axioms of probability theory, conditional probability, random variables, expected value, and major theoretical results of probability. Other subjects include Markov chains, information and coding theory, and simulation. Includes chapter summaries, exercises, and answers. This fifth edition notes optional material, and updates examples to be more accessible to students. Chapter exercises are reorganized to present mechanical problems before theoretical exercises. The disk, new to this edition, allows students to perform calculations and simulations. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

  • ISBN-13: 9780130338518
  • Publisher: Prentice Hall
  • Publication date: 7/1/2001
  • Edition description: Older Edition
  • Edition number: 6
  • Pages: 528
  • Product dimensions: 7.24 (w) x 9.30 (h) x 0.98 (d)

Table of Contents

Preface vi
1 Combinatorial Analysis 1
1.1 Introduction 1
1.2 The Basic Principle of Counting 2
1.3 Permutations 3
1.4 Combinations 5
1.5 Multinomial Coefficients 10
1.6 The Number of Integer Solutions of Equations 12
Summary 15
Problems 15
Theoretical Exercises 18
Self-Test Problems and Exercises 22
2 Axioms of Probability 24
2.1 Introduction 24
2.2 Sample Space and Events 24
2.3 Axioms of Probability 28
2.4 Some Simple Propositions 31
2.5 Sample Spaces Having Equally Likely Outcomes 35
2.6 Probability As a Continuous Set Function 47
2.7 Probability As a Measure of Belief 51
Summary 52
Problems 53
Theoretical Exercises 59
Self-Test Problems and Exercises 61
3 Conditional Probability and Independence 64
3.1 Introduction 64
3.2 Conditional Probabilities 64
3.3 Bayes' Formula 69
3.4 Independent Events 83
3.5 P(-[middle dot]F) is a Probability 96
Summary 103
Problems 104
Theoretical Exercises 115
Self-Test Problems and Exercises 119
4 Random Variables 122
4.1 Random Variables 122
4.2 Discrete Random Variables 127
4.3 Expected Value 130
4.4 Expectatio of a Function of a Random Variable 133
4.5 Variance 137
4.6 The Bernoulli and Binomial Random Variables 139
4.6.1 Properties of Binomial Random Variables 144
4.6.2 Computing the Binomial Distribution Function 147
4.7 The Poisson Random Variable 149
4.7.1 Computing the Poisson Distribution Function 157
4.8 Other Discrete Probability Distribution 158
4.8.1 The Geometric Random Variable 158
4.8.2 The Negative Binomial Random Variable 160
4.8.3 The Hypergeometric Random Variable 162
4.8.4 The Zeta (or Zipf) distribution 166
4.9 Properties of the Cumulative Distribution Function 166
Summary 169
Problems 171
Theoretical Exercises 180
Self-Test Problems and Exercises 184
5 Continuous Random Variables 187
5.1 Introduction 187
5.2 Expectation and Variance of Continuous Random Variables 190
5.3 The Uniform Random Variable 195
5.4 Normal Random Variables 199
5.4.1 The Normal Approximation to the Binomial Distribution 206
5.5 Exponential Random Variables 210
5.5.1 Hazard Rate Functions 215
5.6 Other Continuous Distributions 217
5.6.1 The Gamma Distribution 217
5.6.2 The Weibull Distribution 220
5.6.3 The Cauchy Distribution 220
5.6.4 The Beta Distribution 221
5.7 The Distribution of a Function of a Random Variable 223
Summary 225
Problems 228
Theoretical Exercises 232
Self-Test Problems and Exercises 235
6 Jointly Distributed Random Variables 239
6.1 Joint Distribution Functions 239
6.2 Independent Random Variables 248
6.3 Sums of Independent Random Variables 260
6.4 Conditional Distributions: Discrete Case 268
6.5 Conditional Distributions: Continuous Case 270
6.6 Order Statistics 273
6.7 Joint Probability Distribution of Functions of Random Variables 277
6.8 Exchangeable Random Variables 285
Summary 288
Problems 290
Theoretical Exercises 296
Self-Test Problem and Exercises 299
7 Properties of Expectation 304
7.1 Introduction 304
7.2 Expectation of Sums of Random Variables 305
7.2.1 Obtaining Bounds from Expectations via the Probabilistic Method 321
7.2.2 The Maximum-Minimums Identity 324
7.3 Covariance, Variance of Sums, and Correlations 327
7.4 Conditional Expectation 340
7.4.1 Definitions 340
7.4.2 Computing Expectations by Conditioning 343
7.4.3 Computing Probabilities by Conditioning 350
7.4.4 Conditional Variance 354
7.5 Conditional Expectation and Prediction 356
7.6 Moment Generating Functions 361
7.6.1 Joint Moment Generating Functions 371
7.7 Additional Properties of Normal Random Variables 373
7.7.1 The Multivariate Normal Distribution 373
7.7.2 The Joint Distribution of the Sample Mean and Sample Variance 374
7.8 General Definition of Expectation 375
Summary 377
Problems 379
Theoretical Exercises 389
Self-Test Problems and Exercises 397
8 Limit Theorems 400
8.1 Introduction 400
8.2 Chebyshev's Inequality and the Weak Law of Large Numbers 400
8.3 The Central Limit Theorem 403
8.4 The Strong Law of Large Numbers 412
8.5 Other Inequalities 417
8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson 424
Summary 426
Problems 427
Theoretical Exercises 429
Self-Test Problems and Exercises 430
9 Additional Topics in Probability 432
9.1 The Poisson Process 432
9.2 Markov Chains 435
9.3 Surprise, Uncertainty, and Entropy 440
9.4 Coding Theory and Entropy 445
Summary 451
Theoretical Exercises and Problems 452
Self-Test Problems and Exercises 454
References 454
10 Simulation 455
10.1 Introduction 455
10.2 General Techniques for Simulating Continuous Random Variables 458
10.2.1 The Inverse Transformation Method 458
10.2.2 The Rejection Method 459
10.3 Simulating from Discrete Distributions 465
10.4 Variance Reduction Techniques 467
10.4.1 Use of Antithetic Variables 468
10.4.2 Variance Reduction by Conditioning 468
10.4.3 Control Variates 470
Summary 471
Problems 471
Self-Test Problems and Exercises 474
References 474
Appendix A Answers to Selected Problems 475
Appendix B Solutions to Self-Test Problems and Exercises 478
Index 519
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Preface

"We see that the theory of probability is at bottom only common sense reduced to calculation; it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct, often without being able to account for it .... It is remarkable that this science, which originated in the consideration of games of chance, should have become the most important object of human knowledge .... The most important questions of life are, for the most part, really only problems of probability." So said the famous French mathematician and astronomer (the "Newton of France") Pierre Simon, Marquis de Laplace. Although many people might feel that the famous marquis, who was also one of the great contributors to the development of probability, might have exaggerated somewhat, it is nevertheless true that probability theory has become a tool of fundamental importance to nearly all scientists, engineers, medical practitioners, jurists, and industrialists. In fact, the enlightened individual had learned to ask not "Is it so?" but rather "What is the probability that it is so?"

This book is intended as an elementary introduction to the theory of probability for students in mathematics, statistics, engineering, and the sciences (including computer science, the social sciences and management science) who possess the prerequisite knowledge of elementary calculus. It attempts to present not only the mathematics of probability theory, but also, through numerous examples, the many diverse possible applications of this subject.

In Chapter 1 we present the basic principles of combinatorial analysis, which are most useful in computing probabilities.

In Chapter 2 we consider theaxioms of probability theory and show how they can be applied to compute various probabilities of interest.

Chapter 3 deals with the extremely important subjects of conditional probability and independence of events. By a series of examples we illustrate how conditional probabilities come into play not only when some partial information is available, but also as a tool to enable us to compute probabilities more easily, even when no partial information is present. This extremely important technique of obtaining probabilities by "conditioning" reappears in Chapter 7, where we use it to obtain expectations.

In Chapters 4, 5, and 6 we introduce the concept of random variables. Discrete random variables are dealt with in Chapter 4, continuous random variables in Chapter 5, and jointly distributed random variables in Chapter 6. The important concepts of the expected value and the variance of a random variable are introduced in Chapters 4 and 5: These quantities are then determined for many of the common types of random variables.

Additional properties of the expected value are considered in Chapter 7. Many examples illustrating the usefulness of the result that the expected value of a sum of random variables is equal to the sum of their expected values are presented. Sections on conditional expectation, including its use in prediction, and moment generating functions are contained in this chapter. In addition, the final section introduces the multi-variate normal distribution and presents a simple proof concerning the joint distribution of the sample mean and sample variance of a sample from a normal distribution.

In Chapter 8 we present the major theoretical results of probability theory. In particular, we prove the strong law of large numbers and the central limit theorem. Our proof of the strong law is a relatively simple one which assumes that the random variables have a finite fourth moment, and our proof of the central limit theorem assumes Levy's continuity theorem. Also in this chapter we present such probability inequalities as Markov's inequality, Chebyshev's inequality, and Chernoff bounds. The final section of Chapter 8 gives a bound on the error involved when a probability concerning a sum of independent Bernoulli random variables is approximated by the corresponding probability for a Poisson random variable having the same expected value.

Chapter 9 presents some additional topics, such as Markov chains, the Poisson process, and an introduction to information and coding theory, and Chapter 10 considers simulation.

The sixth edition continues the evolution and fine tuning of the text. There are many new exercises and examples. Among the latter are examples on utility (Example 4c of Chapter 4), on normal approximations (Example 4i of Chapter 5), on applying the lognormal distribution to finance (Example 3d of Chapter 6), and on coupon collecting with general collection probabilities (Example 2v of Chapter 7). There are also new optional subsections in Chapter 7 dealing with the probabilistic method (Subsection 7.2.1), and with the maximum-minimums identity (Subsection 7.2.2).

As in the previous edition, three sets of exercises are given at the end of each chapter. They are designated as Problems, Theoretical Exercises, and Self-Test Problems and Exercises. This last set of exercises, for which complete solutions appear in Appendix B, is designed to help students test their comprehension and study for exams.

Using the website students will be able to perform calculations and simulations quickly and easily in six key areas:

  • Three of the modules derive probabilities for, respectively, binomial, Poisson, and normal random variables.
  • Another module illustrates the central limit theorem. It considers random variables that take on one of the values 0,1, 2, 3, 4 and allows the user to enter the probabilities for these values along with a number n. The module then plots the probability mass function of the sum of n independent random variables of this type. By increasing n one can "see" the mass function converge to the shape of a normal density function.
  • The other two modules illustrate the strong law of large numbers. Again the user enters probabilities for the five possible values of the random variable along with an integer n. The program then uses random numbers to simulate n random variables having the prescribed distribution. The modules graph the number of times each outcome occurs along with the average of all outcomes. The modules differ in how they graph the results of the trials.
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Customer Reviews

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Sort by: Showing all of 10 Customer Reviews

  • Anonymous

    Posted February 20, 2006

    Beginners Beware

    This book should definitely not be titled, 'A First Course in Probability.' In 1999, I graduated from the University of Pennsylvania with a bachelor of science degree in electrical engineering. As part of my curriculum, I took a course in probability theory and earned a grade of A+. I am presently at Rutgers working towards my graduate degree in Mathematics, but they would not accept my transfer credit for the probability course, and so I am taking a probability class for the second time. We happen to be using the book by Ross, and I must say, good luck if you are not an expert in the field of probability. Even with my previous mastery of the subject, I can barely follow some of the examples in the book. This has got to be one of the worst introductory texts that I have ever seen.

    1 out of 1 people found this review helpful.

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  • Anonymous

    Posted December 23, 2008

    Well Done!

    I just finished taking a probability course that used Sheldon Ross' book, and I was very appreciative of the detailed examples and explanations. No short changing on ink and space in this textbook - and this was very good! Also, his solid blend of basic example problems leading to the more complex was helpful in building one's foundation as well as leading to the bigger picture. I must say that I also found having most answers in the back of the book, as well as a self-test section for each chapter with detailed answers very useful in practicing, gaining insight and understanding the topic of probability. I have since bought other Sheldon Ross books for my bookshelf.

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  • Anonymous

    Posted January 3, 2007

    Solid Book

    I used this book in Intro to Probability at South Florida. The book covers all the necessary material, and provides good examples and good problems.

    Was this review helpful? Yes  NoThank you for your feedback.   Report this reviewThank you, this review has been flagged.

  • Anonymous

    Posted November 29, 2004

    An intro for the mathematically inclined

    I found this book very useful in my probability course but the title is misleading if you expect it to supply details or spoon fed answers. It is written as an intro to the subject for math students, not for those who need these skills for another field and thus the theory, although relatively simple, is presented in a manner that assumes the reader has prior experience with proofs. The examples are extremely important and instructional so do not overlook them!

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  • Anonymous

    Posted December 5, 2003

    HORRIBLE HORRIBLE HORRIBLE

    I bought this book as a supplement to the book I was using in my probability class. Although this book did provide SOME, NOT MUCH, but SOME further insight it still lacked proper structure. The book spends much time on theoretical proofs with too little time on applications of the theory. To title the book a 'First Course in Probability' is a fallacy. There are a plethora of assumptions. For example, the book will say...given function xyz, we conclude 123, with out going through the hundreds of steps in between. This is not a book for beginners, nor does it seem like a book for a first course in probability. With the lack for better words, I think the book is grotesquely confusing and absolutely HORRIBLE!! The 1 Star rating doesn't even do the book justice. I would give it 0 Stars, in fact, if there were negative stars, I would provide appropriate deductions.

    Was this review helpful? Yes  NoThank you for your feedback.   Report this reviewThank you, this review has been flagged.

  • Anonymous

    Posted May 20, 2003

    A Comprehensive, Advanced Course in Probability.

    I had this as a college text book at the senior level in Industrial Engineering. If you have a strong math background it's great. Not recommended for math neophytes. Excellent, if you're ready for it.

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  • Anonymous

    Posted December 13, 2002

    NOT a first course in probability

    The book is very difficult to learn from. The text is very dense and the concepts are presented almost exclusively using mathematical proofs with almost no attempt to also provide intuitive explanations. The worst feature is that the proofs and examples, which are the main tools the book uses to teach, skip many steps that the book claims are trivial or obvious. Those steps may be obvious to people who know the subject well, but they weren't obvious to me. In fact, I spent more time tracking down the skipped steps than I did learning the basic material. Another bad feature is that some very important concepts are only discussed in the examples, with no indication in the general text that they're buried close by! In conclusion, read the book as a concise review only after you've already mastered probablility using other sources.

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  • Anonymous

    Posted January 13, 2002

    horrible, for beginners

    This book is the worst introductory book I have ever read. I am a student studying for Course 1 in the actuary field. This is one of the books on the approved study list of the SOA/CAS and we used it in my Introduction to Probability Class. The majority of people in the class never took a class in probability and this book was over everyone's head. The book gave few examples and the study questions at the end of each chapter were so difficult that the teacher stopped assigning the class homework. The book is just not for beginners. It is for those that have a good background in probability and want to go deeper into the field.

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  • Anonymous

    Posted June 4, 2009

    No text was provided for this review.

  • Anonymous

    Posted January 21, 2009

    No text was provided for this review.

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