本书系统阐述了统计模拟的一些实用方法和技术。在对概率的基本知识进行了简单的回顾之后,介绍如何利用计算机产生随机数以及如何利用这些随机数产生任意分布的随机变量、随机过程等。然后讨论了一些分析统计数据的方法和技术,如Bootstrap(自助法)、方差缩减技术等。接着讲述了如何利用统计模拟来判断所选的随机模型是否拟合实际的数据。*后介绍MCMC及一些*新发展的统计模拟技术和论题,如随机序列函数和随机子集函数的评估。本书在每章的*后还提供了不同难度的习题。

1 Introduction 1
Exercises 3
2 Elements of Probability 5
2.1 Sample Space and Events 5
2.2 Axioms of Probability 6
2.3 Conditional Probability and Independence 7
2.4 Random Variables 9
2.5 Expectation 11
2.6 Variance 14
2.7 Chebyshev's Inequality and the Laws of Large Numbers 16
2.8 Some Discrete Random Variables 18
Binomial Random Variables 18
Poisson Random Variables 20
Geometric Random Variables 22
The Negative Binomial Random Variable 23
Hypergeometric Random Variables 24
2.9 Continuous Random Variables 24
Uniformly Distributed Random Variables 25
Normal Random Variables 26
Exponential Random Variables 27
The Poisson Process and Gamma Random Variables 29
The Nonhomogeneous Poisson Process 32
2.10 Conditional Expectation and Conditional Variance 33
The Conditional Variance Formula 34
Exercises 35
References 39
3 Random Numbers 41
Introduction 41
3.1 Pseudorandom Number Generation 41
3.2 Using Random Numbers to Evaluate Integrals 42
Exercises 46
References 48
4 Generating Discrete Random Variables 49
4.1 The Inverse Transform Method 49
4.2 Generating a Poisson Random Variable 55
4.3 Generating Binomial Random Variables 57
4.4 The Acceptance-Rejection Technique 58
4.5 The Composition Approach 60
4.6 Generating Random Vectors 61
Exercises 62
5 Generating Continuous Random Variables 67
Introduction 67
5.1 The Inverse Transform Algorithm 67
5.2 The Rejection Method 71
5.3 The Polar Method for Generating Normal Random Variables 78
5.4 Generating a Poisson Process 82
5.5 Generating a Nonhomogeneous Poisson Process 83
Exercises 87
References 91
6 The Discrete Event Simulation Approach 93
Introduction 93
6.1 Simulation via Discrete Events 93
6.2 A Single-Server Queueing System 94
6.3 A Queueing System with Two Servers in Series 97
6.4 A Queueing System with Two Parallel Servers 99
6.5 An Inventory Model 102
6.6 An Insurance Risk Model 103
6.7 A Repair Problem 105
6.8 Exercising a Stock Option 108
6.9 Verification of the Simulation Model 110
Exercises 111
References 115
7 Statistical Analysis of Simulated Data 117
Introduction 117
7.1 The Sample Mean and Sample Variance 117
7.2 Interval Estimates of a Population Mean 123
7.3 The Bootstrapping Technique for Estimating Mean Square Errors 126
Exercises 133
References 135
8 Variance Reduction Techniques 137
Introduction 137
8.1 The Use of Antithetic Variables 139
8.2 The Use of Control Variates 147
8.3 Variance Reduction by Conditioning 154
Estimating the Expected Number of Renewals by Time t 164
8.4 Stratified Sampling 166
8.5 Applications of Stratified Sampling 175
Analyzing Systems Having Poisson Arrivals 176
Computing Multidimensional Integrals of Monotone Functions 180
Compound Random Vectors 182
8.6 Importance Sampling 184
8.7 Using Common Random Numbers 197
8.8 Evaluating an Exotic Option 198
8.9 Estimating Functions of Random Permutations and Random Subsets 203
Random Permutations 203
Random Subsets 206
8.10 Appendix: Verification of Antithetic Variable Approach When Estimating the Expected Value of Monotone Functions 207
Exercises 209
References 217
9 Statistical Validation Techniques 219
Introduction 219
9.1 Goodness of Fit Tests 219
The Chi-Square Goodness of Fit Test for Discrete Data 220
The Kolmogorov-Smirnov Test for Continuous Data 222
9.2 Goodness of Fit Tests When Some Parameters Are Unspecified 227
The Discrete Data Case 227
The Continuous Data Case 230
9.3 The Two-Sample Problem 230
9.4 Validating the Assumption of a Nonhomogeneous Poisson Process 237
Exercises 241
References 244
10 Markov Chain Monte Carlo Methods 245
Introduction 245
10.1 Markov Chains 245
10.2 The Hastings-Metropolis Algorithm 248
10.3 The Gibbs Sampler 251
10.4 Simulated Annealing 262
10.5 The Sampling Importance Resampling Algorithm 264
Exercises 269
References 272
11 Some Additional Topics 273
Introduction 273
11.1 The Alias Method for Generating Discrete Random Variables 273
11.2 Simulating a Two-Dimensional Poisson Process 277
11.3 Simulation Applications of an Identity for Sums of Bernoulli Random Variables 280
11.4 Estimating the Distribution and the Mean of the First Passage Time of a Markov Chain 285
11.5 Coupling from the Past 289
Exercises 291
References 293
Index 294

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