**Introduction **

“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 feel that the famous marquis, who was also one of the great contributors to the development of probability, might have exaggerated somewhat, it is never the less 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, biology, 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.

Chapter 1 presents the basic principles of combinatorial analysis, which are most useful in computing probabilities. Chapter 2 handles the axioms of probability theory and shows 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.

The concept of random variables is introduced in Chapters 4, 5, and 6. Discrete random variables are dealt with in Chapter 4, continuous random variables in Chapter 5,andjointlydistributedrandomvariablesinChapter 6.Theimportantconcepts of the expected value and the variance of a random variable are introduced in Chapters 4 and 5, and 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 it sasin prediction, and on moment-generating functions are contained in this chapter. In addition, the ﬁnal section introduces the multivariate 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.

Chapter 8 presents the major theoretical results of probability theory. In particular, we prove the strong law of large numbers and the central limit theorem. Our proofofthestronglawisarelativelysimpleonewhichassumesthattherandomvariableshaveaﬁnitefourthmoment,and our proof of the central limit theorem assumes Levy’scontinuitytheorem.Thischapteralsopresentssuchprobabilityinequalitiesas 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 of a Poisson random variable having the same expected value. Chapter 9presentssomeadditionaltopics, such as Markov chains, the Poisson process, and an introduction to information and coding theory, and Chapter 10 considers simulation.

The eighth edition continues the evolution and ﬁne tuning of the text. It includes new problems, exercises, and text material chosen both for its inherent interest and for its use in building student intuition about probability. Illustrative of these goals are Example 5d of Chapter 1 on knockout tournaments, and Examples 4k and 5i of Chapter 7 on multiple player gambler’s ruin problems. Akeychangeinthecurrenteditionisthattheimportantresultthattheexpectation of a sum of random variables is equal to the sum of the expectations is now ﬁrst presented in Chapter 4 (rather than Chapter 7 as in previous editions). A new and elementary proof of this result when the sample space of the probability experiment is ﬁnite is given in this chapter. Another change is the expansion of Section 6.3, which deals with the sum of independent random variables. Section 6.3.1 is a new section in which we derive the distribution of the sum of independent and identically distributed uniform random variables, and then use our results to show that the expected number of random numbers that needs to be added for their sum to exceed 1 is equal to e. Section 6.3.5 is a new section in which we derive the distribution of the sum of independent geometric random variables with different means

**Contents **

**1 Combination Analysis 1**

1.1 Introduction

1.2 The Basic Principle of Counting

1.3 Permutations

1.4 Combinations

1.5 Multinomial Coefficients

1.6 The Number of Integer Solutions of Equations

Summary

Problems

Theoretical Exercises

Self-Test Problems and Exercises

**2 Axioms of Probability 22**

2.1 Introduction

2.2 Sample Space and Events

2.3 Axioms of Probability

2.4 Some Simple Propositions.

2.5 Sample Spaces Having Equally Likely Outcomes.

2.6 Probability as a Continuous Set Function

2.7 Probability as a Measure of Belief

Summary

Problems

Theoretical Exercises

Self-Test Problems and Exercises

**3 Conditional Probability and Independence 58**

3.1 Introduction

3.2 Conditional Probabilities

3.3 Bayles’s Formula

3.4 Independent Events

3.5 *P*(**·**|*F*) Is a Probability

Summary

Problems

Theoretical Exercises

Self-Test Problems and Exercises

**4 Random Variables 117**

4.1 Random Variables

4.2 Discrete Random Variables

4.3 Expected Value

4.4 Expectation of a Function of a Random Variable

4.5 Variance

4.6 The Bernoulli and Binomial Random Variables

4.6.1 Properties of Binomial Random Variables.

4.6.2 Computing the Binomial Distribution Function

4.7 The Poisson Random Variable

4.7.1 Computing the Poisson distribution Function

4.8 Other Discrete Probability Distributions

4.8.1 The Geometric Random Variable

4.8.2 The Negative Binomial Random Variable

4.8.3 The Hypergeometric Random Variable

4.8.4 The Zeta (orZipf) Distribution

4.9 Expected Value of Sums of Random Variables

4.10 Properties of the Cumulative Distribution Function

Summary

Problems.

Theoretical Exercises

Self-Test Problems and Exercises

**5 Continuous Random Variables 186**

5.1 Introduction

5.2 Expectation and Variance of Continuous Random Variables

5.3 The Uniform Random Variable

5.4 Normal Random Variables

5.4.1 The Normal Approximation to the Binomial Distribution

5.5 Exponential Random Variables

5.5.1 Hazard Rate Functions

5.6 Other Continuous Distributions

5.6.1 The Gamma Distribution

5.6.2 The Weibull Distribution

5.6.3 The Cauchy Distribution

5.6.4 The Beta Distribution

5.7 The Distribution of a Function of a Random Variable

Summary

Problems

Theoretical Exercises

Self-Test Problems and Exercises

**6 Jointly Distributed Random Variables 232**

6.1 Joint Distribution Functions

6.2 Independent Random Variables

6.3 Sums of Independent Random Variables

6.3.1 Identically Distributed Uniform Random Variables

6.3.2 Gamma Random Variables

6.3.3 Normal Random Variables

6.3.4 Poisson and Binomial Random Variables

6.3.5 Geometric Random Variables

6.4 Conditional Distributions: Discrete Case

6.5 Conditional Distributions: Continuous Case

6.6 Order Statistics

6.7 Joint Probability Distribution of Functions of Random Variables

6.8 Exchangeable Random Variables

Summary

Problems

Theoretical Exercises

Self-Test Problems and Exercises

**7 Properties of Expectation 297**

7.1 Introduction

7.2 Expectation of Sums of Random Variables

7.2.1 Obtaining Bounds from Expectations

via the Probabilistic Method

7.2.2 The Maximum–Minimums Identity

7.3 Moments of the Number of Events that Occur

7.4 Covariance, Variance of Sums, and Correlations

7.5 Conditional Expectation

7.5.1 Definitions

7.5.2 Computing Expectations by Conditioning

7.5.3 Computing Probabilities by Conditioning

7.5.4 Conditional Variance

7.6 Conditional Expectation and Prediction

7.7 Moment Generating Functions

7.7.1 Joint Moment Generating Functions

7.8 Additional Properties of Normal Random Variables

7.8.1 The Multivariate Normal Distribution

7.8.2 The Joint Distribution of the Sample Mean And Sample Variance

7.9 General Definition of Expectation

Summary

Problems.

Theoretical Exercises

Self-Test Problems and Exercises

**8 Limit Theorems 388**

8.1 Introduction

8.2 Chebyshev’s Inequality and the Weak Law of Large

Numbers

8.3 The Central Limit Theorem.

8.4 The Strong Law of Large Numbers

8.5 Other Inequalities.

8.6 Bounding the Error Probability When Approximating a Sum of

Independent Bernoulli Random Variables by a Poisson

Random Variable.

Summary.

Problems

Theoretical Exercises

Self-Test Problems and Exercises

**9 Additional Topics in Probability 417**

9.1 The Poisson Process

9.2 Markov Chains . .

9.3 Surprise, Uncertainty, and Entropy

9.4 Coding Theory and Entropy .

Summary .

Problems and Theoretical Exercises

Self-Test Problems and Exercises

References .

**10 Simulation 438**

10.1 Introduction

10.2 General Techniques for Simulating Continuous Random Variables

10.2.1 The Inverse Transformation Method

10.2.2 The Rejection Method

10.3 Simulating from Discrete Distributions

10.4 Variance Reduction Techniques

10.4.1 Use of Antithetic Variables

10.4.2 Variance Reduction by Conditioning

10.4.3 Control Variates

Summary

Problems

Self-Test Problems and Exercises

Reference

**Answers to Selected Problems 457**

**Solutions to Self-Test Problems and Exercises 461**

**Index**

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