应用随机过程：概率模型导论（英文影印版·第8版）

.2.3.3. gamma random variables 37
2.3.4. normal random variables 37
2.4. expectation of a random variable 38
2.4.1. the discrete case 38
2.4.2. the continuous case 41
2.4.3. expectation of a function of a random variable 43
2.5. jointly distributed random variables 47
2.5.1. joint distribution functions 47
2.5.2. independent random variables 51
2.5.3. covariance and variance of sums of random variables 53
2.5.4. joint probability distribution of functions of random variables 61
2.6. moment generating functions 64
2.6.1. the joint distribution of the sample mean and sample variance from a normal population 74
2.7. limit theorems 77
2.8. stochastic processes 83
exercises 85
references 96
3. conditional probability and conditional expectation 97
3.1. introduction 97
3.2. the discrete case 97
3.3. the continuous case 102
3.4. computing expectations by conditioning 105
3.4.1. computing variances by conditioning 116
3.5. computing probabilities by conditioning 119
3.6. some applications 136
3.6.1. a list model 136
3.6.2. arandom graph 138
3.6.3. uniform priors, polya's urn model, and bose-einstein statistics 146
3.6.4. mean time for patterns 150
3.6.5. a compound poisson identity 154
3.6.6. the k-record values of discrete random variables 158
exercises 161
4. markov chains 181
4.1. introduction 181
4.2. chapman-kolmogorov equations 185
4.3. classification of states 189
4.4. limiting probabilities 200
4.5. some applications 213
4.5.1. the gambler's ruin problem 213
4.5.2. a model for algorithmic efficiency 217
4.5.3. using a random walk to analyze a probabilistic algorithm for the satisfiability problem 220
4.6. mean time spent in transient states 226
4.7. branching processes 228
4.8. time reversible markov chains 232
4.9. markov chain monte carlo methods 243
4.10. markov decision processes 248
exercises 252
references 268
5. the exponential distribution and the poisson process 269
5.1. introduction 269
5.2. the exponential distribution 270
5.2.1. definition 270
5.2.2. properties of the exponential distribution 272
5.2.3. further properties of the exponential distribution 279
5.2.4. convolutions of exponential random variables 284
5.3. the poisson process 288
5.3.1. counting processes 288
5.3.2. definition of the poisson process 289
5.3.3. interarrival and waiting time distributions 293
5.3.4. further properties of poisson processes 295
5.3.5. conditional distribution of the arrival times 301
5.3.6. estimating software reliability 313
5.4. generalizations of the poisson process 316
5.4.1. nonhomogeneous poisson process 316
5.4.2. compound poisson process 321
5.4.3. conditional or mixed poisson processes 327
exercises 330
references 348
6. continuous-time markov chains 349
6.1. introduction 349
6.2. continuous-time markov chains 350
6.3. birth and death processes 352
6.4. the transition probability function pij (t) 359
6.5. limiting probabilities 368
6.6. time reversibility 376
6.7. uniformization 384
6.8. computing the transition probabilities 388
exercises 390
references.. 399
7. renewal theory and its applications 401
7.1. introduction 401
7.2. distribution of n(t) 403
7.3. limit theorems and their applications 407
7.4. renewal reward processes 416
7.5. regenerative processes 425
7.5.1. alternating renewal processes 428
7.6. semi-markov processes 434
7.7. the inspection paradox 437
7.8. computing the renewal function 440
7.9. applications to patterns 443
7.9.1. patterns of discrete random variables 443
7.9.2. the expected time to a maximal run of distinct values 451
7.9.3. increasing runs of continuous random variables 453
7.10. the insurance ruin problem 455
exercises 460
references 472
8. queueing theory 475
8.1. introduction 475
8.2. preliminaries 476
8.2.1. cost equations 477
8.2.2. steady-state probabilities 478
8.3. exponential models 480
8.3.1. a single-server exponential queueing system 480
8.3.2. a single-server exponential queueing system having finite capacity 487
8.3.3. a shoeshine shop 490
8.3.4. a queueing system with bulk service 493
8.4. network of queues 496
8.4.1. open systems 496
8.4.2. closed systems 501
8.5. the system m/g/1 507
8.5.1. preliminaries: work and another cost identity 507
8.5.2. application of work to m/g/1 508
8.5.3. busy periods 509
8.6. variations on the m/g/1 510
8.6.1. the m/g/1 with random-sized batch arrivals 510
8.6.2. priority queues 512
8.6.3. an m/g/1 optimization example 515
8.7. the model g/m/1 519
8.7.1. the g/m/1 busy and idle periods 524
8.8. a finite source model 525
8.9. multiserver queues 528
8.9.1. erlang's loss system 529
8.9.2. the m/m/k queue 530
8.9.3. the g/m/k queue 530
8.9.4. the m/g/k queue 532
exercises 534
references 546
9. reliability theory 547
9.1. introduction 547
9.2. structure functions 547
9.2.1. minimal path and minimal cut sets 550
9.3. reliability of systems of independent components 554
9.4. bounds on the reliability function 559
9.4.1. method of inclusion and exclusion 560
9.4.2. second method for obtaining bounds on r(p) 569
9.5. system life as a function of component lives 571
9.6. expected system lifetime 580
9.6.1. an upper bound on the expected life of a parallel system 584
9.7. systems with repair 586
9.7.1. a series model with suspended animation 591
exercises 593
references 600
10. brownian motion and stationary processes 601
10.1. brownian motion 601
10.2. hitting times, maximum variable, and the gambler's ruin problem 605
10.3. variations on brownian motion 607
10.3.1. brownian motion with drift 607
10.3.2. geometric brownian motion 607
10.4. pricing stock options 608
10.4.1. an example in options pricing 608
10.4.2. the arbitrage theorem 611
10.4.3. the black-scholes option pricing formula 614
10.5. white noise 620
10.6. ganssian processes 622
10.7. stationary and weakly stationary processes 625
10.8. harmonic analysis of weakly stationary processes 630
exercises 633
references 638
11. simulation 639
11.1. introduction 639
11.2. general techniques for simulating continuous random variables 644
11.2.1. the inverse transformation method 644
11.2.2. the rejection method 645
11.2.3. the hazard rate method 649
11.3. special techniques for simulating continuous random variables 653
11.3.1. the normal distribution 653
11.3.2. the gamma distribution 656
11.3.3. the chi-squared distribution 657
11.3.4. the beta (n, m) distribution 657
11.3.5. the exponential distribution--the von neumann algorithm 658
11.4. simulating from discrete distributions 661
11.4.1. the alias method 664
11.5. stochastic processes 668
11.5.1. simulating a nonhomogeneous poisson process 669
11.5.2. simulating a two-dimensional poisson process 676
11.6. variance reduction techniques 679
11.6.1. use of antithetic variables 680
11.6.2. variance reduction by conditioning 684
11.6.3. controlvariates 688
11.6.4. importance sampling 690
11.7. determining the number of runs 696
11.8. coupling from the past 696
exercises 699
references 707
appendix: solutions to starred exercises 709
index... 749

.　　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. Section 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 theory, and it is stressed throughout the book. When properly used, conditioning often 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. Section 3.6.6 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 presented. The concept of time reversibility is introduced and its usefulness illustrated. Section 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 exponential distribution is discussed. New derivations for the Poisson and nonhomogeneous 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 in this chapter. Section 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 computationally important technique of uniformization.
　　Chapter 7, the renewal theory chapter, is concerned with a type of counting 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 occurs when a sequence of independent and identically distributed random variables is observed. In Section 7.9.1, we show how renewal theory can be used to derive 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 Section 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 Section 7.9.3, we suppose the random variables are continuous and derive an expression for the mean time until a run ofm consecutive increasing values occurs.
　　Chapter 8 deals with queueing, or waiting line, theory. After some preliminaries dealing with basic cost identities and types of limiting probabilities, we consider 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. Section 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 models that are analytically intractable. Methods for generating the values of arbitrarily distributed random variables are discussed, as are variance reduction methods for increasing the efficiency of the simulation. Section 11.6.4 introduces the important simulation technique of importance sampling, and indicates the usefulness of tilted distributions when applying this method. ...