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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Book Description
Editorial Reviews
From the Back Cover
Excerpt. © Reprinted by permission. All rights reserved.
"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 the axioms 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.
All materials included on the Probability Models diskette from previous editions can now be downloaded from the Ross companion website at http://www.prenhall.com/ross. 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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28 of 30 people found the following review helpful:
5.0 out of 5 stars
A Classic of Probability Theory,
By
This review is from: A First Course in Probability (6th Edition) (Hardcover)
A First Course in Probability by Sheldon Ross covers all the main topics of probability theory: Combinatorics, Probability Axioms, Conditional Probability and Independence, Discrete Random Variables, Continuous Random Variables, Joint Distributions, Expectation, and Limit Theorems. He develops each topic thoroughly using the definition-theorem-proof approach of classical mathematics, interspersed with numerous examples, many of which are classics in probability.
This book does require a solid foundation in calculus. Consequently, it is an appropriate text for a course at an advanced undergraduate level or even a first year graduate course (which is where I first encountered it). It does not require any knowledge of truly advanced mathematics (i.e., measure theory) which one would expect to find in an upper level graduate text, such as Patrick Billingsley's Probability and Measure. Advice to students (and teachers): A student who does not have a solid foundation in calculus, as evidenced by the ability to apply integration by parts, and perhaps a year of post-calculus math which introduced the concept of the mathematical proof, will have a difficult time with this book. This book provided me with all the probability theory I needed to complete a master's degree in statistics. Since statistics is nothing more than a collection of applied problems that can be solved, modeled, or at least understood by using the tools of probability theory, I was able to coast through the rest of my master's program and didn't have to start really working again until I subsequently encountered Billingsley's book (cited above). Thank you, Professor Ross.
12 of 12 people found the following review helpful:
4.0 out of 5 stars
Suitable preparation for Actuary Exam P,
By
This review is from: A First Course in Probability (6th Edition) (Hardcover)
I used this book (as well as the SOA sample exams) as my sole means of preparation for the Actuary Examination P. Ross does an excellent job of utilizing a plethora of examples in order teach a particular concept. In some cases, a chapter may consist of two pages of instruction, followed by dozens of examples. This method tends to be extremely effective with regard to test preparation, but more frustrating for students using this as a course textbook. The proofs provided are fairly vigorous, but a strong background in Calculus is essential to understanding any significant portion of this book.
The chapter discussing "Jointly Distributed Random Variables" is the longest in the book, and perhaps rightly so, considering this comprises the majority of the P exam. However, in 50 pages of text, about five pages involves a formal discussion of terms and proofs, and the rest is made up of about 40 examples, many which span multiple pages. Again, this happens to be quite effective when paired with the SOA example questions, but may prove to be a difficult text to follow when a professor expects you to understand the finer points of probability theory. If you can answer every question provided in a sample exam packet, you will easily pass the exam; and you will master the sample exam if you spend several months covering the first seven chapters of this book.
6 of 6 people found the following review helpful:
5.0 out of 5 stars
Fantastic Introductory Book,
By
This review is from: A First Course in Probability (6th Edition) (Hardcover)
I used this book in my second year of undergrad in an introductory probability course. This book was perfect for such a course. It is a very easy introduction to the world of probability. Ross assumes almost nothing but basic calculus and a willingness to work through the problems to increase your understanding. Some of the other reviews I have read basically are asking for the author to hold your hand and do all the work for you. I felt like this book actually does this a little too much at times, but really strikes an almost perfect balance between simplicity and technical rigor.
Obviously one needs to move on to much more technical and rigorous texts before one actually has an understanding of probability. However, one of the most important abilities I have found in my academic life is the ability to use the basics that I learned in Ross to conceptualize much more difficult concepts. I continually refer back to this book for a intuitive explanation of certain basic concepts when I feel a little rusty during my every day use of probability. One must be aware though that as my current probability teacher said, probability is inherently challenging, it is not as trivial as calculus. In the sense that our natural world, at a first approximation, follows the laws of Newtonian physics for which calculus was invented; however, probability is much less intuitive and everyone has a difficulty with it at some point or another.
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