《应用随机过程概率模型导论》Sheldon M. Ross 9版概览

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"应用随机过程概率模型导论 9版 - Sheldon M. Ross" 《应用随机过程概率模型导论》是Sheldon M. Ross撰写的一本经典教材，现在已经更新到了第九版。这本书是概率论和随机过程领域的基石读物，旨在为学生和专业人员提供深入理解和应用这些理论的基础。在本书中，作者Sheldon M. Ross，一位来自加州大学伯克利分校的教授，以清晰易懂的方式介绍了概率模型和随机过程的基本概念。他将理论与实际问题相结合，使读者能够更好地理解这些抽象概念在现实生活和工程中的应用。第九版的内容可能包括了对之前版本的更新和改进，确保与最新的研究成果和应用保持同步。书中的章节可能涵盖了： 1. 概率基础：概率的定义、古典概率、条件概率、贝叶斯定理、独立事件等基础知识。 2. 随机变量：离散随机变量、连续随机变量、期望值、方差、概率分布函数和累积分布函数。 3. 大数定律与中心极限定理：这两个定理是概率论的核心，解释了大量独立随机变量求和后的行为。 4. 马尔科夫链：一种重要的随机过程，描述状态之间的转移概率，广泛应用于排队论、生物统计学和金融等领域。 5. 泊松过程：用于模拟事件发生时间的随机过程，常见于保险、通信和物理学。 6. 跳跃过程和布朗运动：这些过程在物理学、化学和金融建模中扮演着重要角色。 7. 二项式过程、泊松过程的分支结构和其他随机过程：这些高级主题提供了更复杂系统行为的数学描述。 8. 应用实例：书中可能包含各种实际案例，如排队系统、随机搜索、风险理论和随机控制问题。此外，每个章节很可能配备了习题和例题，帮助读者巩固理解并提升解决问题的能力。书的末尾可能还有答案和提示，以及进一步阅读的参考文献，便于深入研究。《应用随机过程概率模型导论》第九版是一部全面而权威的概率论和随机过程教科书，适合统计学、工程学、计算机科学、经济学等相关专业的学生和研究人员学习使用。通过学习这本书，读者不仅可以掌握概率论的基础知识，还能了解到如何将这些理论应用于解决实际问题。

Preface xv

Examples and Exercises

Many examples are worked out throughout the text, and there are also a large

number of exercises to be solved by students. More than 100 of these exercises

have been starred and their solutions provided at the end of the text. These starred

problems can be used for independent study and test preparation. An Instructor’s

Manual, containing solutions to all exercises, is available free to instructors who

adopt the book for class.

Organization

Chapters 1 and 2 deal with basic ideas of probability theory. In Chapter 1 an

axiomatic framework is presented, while in Chapter 2 the important concept of

a random variable is introduced. Subsection 2.6.1 gives a simple derivation of

the joint distribution of the sample mean and sample variance of a normal data

sample.

Chapter 3 is concerned with the subject matter of conditional probability and

conditional expectation. “Conditioning” is one of the key tools of probability the-

ory, and it is stressed throughout the book. When properly used, conditioning of-

ten enables us to easily solve problems that at ﬁrst glance seem quite difﬁcult. The

ﬁnal section of this chapter presents applications to (1) a computer list problem,

(2) a random graph, and (3) the Polya urn model and its relation to the Bose-

Einstein distribution. Subsection 3.6.5 presents k-record values and the surprising

Ignatov’s theorem.

In Chapter 4 we come into contact with our ﬁrst random, or stochastic, process,

known as a Markov chain, which is widely applicable to the study of many real-

world phenomena. Applications to genetics and production processes are pre-

sented. The concept of time reversibility is introduced and its usefulness illus-

trated. Subsection 4.5.3 presents an analysis, based on random walk theory, of a

probabilistic algorithm for the satisﬁability problem. Section 4.6 deals with the

mean times spent in transient states by a Markov chain. Section 4.9 introduces

Markov chain Monte Carlo methods. In the ﬁnal section we consider a model for

optimally making decisions known as a Markovian decision process.

In Chapter 5 we are concerned with a type of stochastic process known as a

counting process. In particular, we study a kind of counting process known as

a Poisson process. The intimate relationship between this process and the expo-

nential distribution is discussed. New derivations for the Poisson and nonhomo-

geneous Poisson processes are discussed. Examples relating to analyzing greedy

algorithms, minimizing highway encounters, collecting coupons, and tracking the

AIDS virus, as well as material on compound Poisson processes, are included

xvi Preface

in this chapter. Subsection 5.2.4 gives a simple derivation of the convolution of

exponential random variables.

Chapter 6 considers Markov chains in continuous time with an emphasis on

birth and death models. Time reversibility is shown to be a useful concept, as it

is in the study of discrete-time Markov chains. Section 6.7 presents the computa-

tionally important technique of uniformization.

Chapter 7, the renewal theory chapter, is concerned with a type of count-

ing process more general than the Poisson. By making use of renewal reward

processes, limiting results are obtained and applied to various ﬁelds. Section 7.9

presents new results concerning the distribution of time until a certain pattern oc-

curs when a sequence of independent and identically distributed random variables

is observed. In Subsection 7.9.1, we show how renewal theory can be used to de-

rive both the mean and the variance of the length of time until a speciﬁed pattern

appears, as well as the mean time until one of a ﬁnite number of speciﬁed patterns

appears. In Subsection 7.9.2, we suppose that the random variables are equally

likely to take on any of m possible values, and compute an expression for the

mean time until a run of m distinct values occurs. In Subsection 7.9.3, we sup-

pose the random variables are continuous and derive an expression for the mean

time until a run of m consecutive increasing values occurs.

Chapter 8 deals with queueing, or waiting line, theory. After some preliminar-

ies dealing with basic cost identities and types of limiting probabilities, we con-

sider exponential queueing models and show how such models can be analyzed.

Included in the models we study is the important class known as a network of

queues. We then study models in which some of the distributions are allowed to

be arbitrary. Included are Subsection 8.6.3 dealing with an optimization problem

concerning a single server, general service time queue, and Section 8.8, concerned

with a single server, general service time queue in which the arrival source is a

ﬁnite number of potential users.

Chapter 9 is concerned with reliability theory. This chapter will probably be

of greatest interest to the engineer and operations researcher. Subsection 9.6.1

illustrates a method for determining an upper bound for the expected life of a

parallel system of not necessarily independent components and (9.7.1) analyzing

a series structure reliability model in which components enter a state of suspended

animation when one of their cohorts fails.

Chapter 10 is concerned with Brownian motion and its applications. The theory

of options pricing is discussed. Also, the arbitrage theorem is presented and its

relationship to the duality theorem of linear program is indicated. We show how

the arbitrage theorem leads to the Black–Scholes option pricing formula.

Chapter 11 deals with simulation, a powerful tool for analyzing stochastic mod-

els that are analytically intractable. Methods for generating the values of arbitrar-

ily distributed random variables are discussed, as are variance reduction methods

for increasing the efﬁciency of the simulation. Subsection 11.6.4 introduces the

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