Preface
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
2.9 Continuous Random Variables 23
2.10 Conditional Expectation and Conditional Variance 31
Exercises 33
Bibliography 38
3 Random Numbers 39
Introduction 39
3.1 Pseudorandom Number Generation 39
3.2 Using Random Numbers to Evaluate Integrals 40
Exercises 44
Bibliography 45
4 Generating Discrete Random Variables 47
4.1 The Inverse Transform Method 47
4.2 Generating a Poisson Random Variable 54
4.3 Generating Binomial Random Variables 55
4.4 The Acceptance-Rejection Technique 56
4.5 The Composition Approach 58
4.6 The Alias Method for Generating Discrete Random Variables 60
4.7 Generating Random Vectors 63
Exercises 64
5 Generating Continuous Random Variables 69
Introduction 69
5.1 The Inverse Transform Algorithm 69
5.2 The Rejection Method 73
5.3 The Polar Method for Generating Normal Random Variables 80
5.4 Generating a Poisson Process 83
5.5 Generating a Nonhomogeneous Poisson Process 85
5.6 Simulating a Two-Dimensional Poisson Process 88
Exercises 91
Bibliography 95
6 The Multivariate Normal Distribution and Copulas 97
Introduction 97
6.1 The Multivariate Normal 97
6.2 Generating a Multivariate Normal Random Vector 99
6.3 Copulas 102
6.4 Generating Variables from Copula Models 107
Exercises 108
7 The Discrete Event Simulation Approach 11 !
Introduction 111
7.1 Simulation via Discrete Events 111
7.2 A Single-Server Queueing System 112
7.3 A Queueing System with Two Servers in Series 115
7.4 A Queueing System with Two Parallel Servers 117
7.5 An Inventory Model 120
7.6 An Insurance Risk Model 122
7.7 A Repair Problem 124
7.8 Exercising a Stock Option 126
7.9 Verification of the Simulation Model 128
Exercises 129
Bibliography 134
8 Statistical Analysis of Simulated Data 135
Introduction 135
8.1 The Sample Mean and Sample Variance 135
8.2 Interval Estimates of a Population Mean 141
8.3 The Bootstrapping Technique for Estimating Mean Square Errors 144
Exercises 150
Bibliography 152
9 Variance Reduction Techniques 153
Introduction 153
9.1 The Use of Antithetic Variables 155
9.2 The Use of Control Variates 162
9.3 Variance Reduction by Conditioning 169
9.4 Stratified Sampling 182
9.5 Applications of Stratified Sampling 192
9.6 Importance Sampling 201
9.7 Using Common Random Numbers 214
9.8 Evaluating an Exotic Option 216
9.9 Appendix: Verification of Antithetic Variable Approach When Estimating the Expected Value of Monotone Functions 220
Exercises 222
Bibliography 231
10 Additional Variance Reduction Techniques 233
Introduction 233
10.1 The Conditional Bernoulli Sampling Method 233
10.2 Normalized Importance Sampling 240
10.3 Latin Hypercube Sampling 244
Exercises 246
11 Statistical Validation Techniques 247
Introduction 247
11.1 Goodness of Fit Tests 247
11.2 Goodness of Fit Tests When Some Parameters Are Unspecified 254
11.3 The Two-Sample Problem 257
11.4 Validating the Assumption of a Nonhomogeneous Poisson Process 263
Exercises 267
Bibliography 270
12 Markov Chain Monte Carlo Methods 271
Introduction 271
12.1 Markov Chains 271
12.2 The Hastings-Metropolis Algorithm 274
12.3 The Gibbs Sampler 276
12.4 Continuous time Markov Chains and a Queueing Loss Model 287
12.5 Simulated Annealing 290
12.6 The Sampling Importance Resampling Algorithm 293
12.7 Coupling from the Past 297
Exercises 298
Bibliography 301
Index 303
