A First Course in Probability / Edition 6

by Sheldon Ross

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.

Product Details

ISBN-13:	9780130338518
Publisher:	Prentice Hall
Publication date:	07/01/2001
Edition description:	Older Edition
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

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.

Editorial Reviews

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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