Simulation

Front Cover
Simulation
Copyright
Contents
Preface
	Overview
	New to this edition
	Chapter descriptions
	Thanks
1 Introduction
	Exercises
2 Elements of probability
	2.1 Sample space and events
	2.2 Axioms of probability
	2.3 Conditional probability and independence
	2.4 Random variables
	2.5 Expectation
	2.6 Variance
	2.7 Chebyshev's inequality and the laws of large numbers
	2.8 Some discrete random variables
		Binomial random variables
		Poisson random variables
		Geometric random variables
		The negative binomial random variable
		Hypergeometric random variables
	2.9 Continuous random variables
		Uniformly distributed random variables
		Normal random variables
		Exponential random variables
		The Poisson process and gamma random variables
		The nonhomogeneous Poisson process
	2.10 Conditional expectation and conditional variance
		The conditional variance formula
	Exercises
	References
3 Random numbers
	Introduction
	3.1 Pseudorandom number generation
	3.2 Using random numbers to evaluate integrals
	Exercises
	References
4 Generating discrete random variables
	4.1 The inverse transform method
	4.2 Generating a Poisson random variable
	4.3 Generating binomial random variables
		Inverse transform algorithm for generating a binomial (n, p) random variable
	4.4 The acceptance–rejection technique
		Rejection method
	4.5 The composition approach
	4.6 The alias method for generating discrete random variables
	4.7 Generating random vectors
	Exercises
5 Generating continuous random variables
	Introduction
	5.1 The inverse transform algorithm
	5.2 The rejection method
		The rejection method
		5.2.1 Order statistics and beta random variables
	5.3 The polar method for generating normal random variables
	5.4 Generating a Poisson process
		Generating the first T time units of a Poisson process with rate λ
	5.5 Generating a nonhomogeneous Poisson process
		Generating the first T time units of a nonhomogeneous Poisson process
		Generating the first T time units of a nonhomogeneous Poisson process
	5.6 Simulating a two-dimensional Poisson process
	Exercises
	References
6 The multivariate normal distribution and copulas
	Introduction
	6.1 The multivariate normal
	6.2 Generating a multivariate normal random vector
	6.3 Copulas
		Multidimensional copulas
	6.4 Generating variables from copula models
	Exercises
7 The discrete event simulation approach
	Introduction
	7.1 Simulation via discrete events
	7.2 A single-server queueing system
	7.3 A queueing system with two servers in series
	7.4 A queueing system with two parallel servers
	7.5 An inventory model
	7.6 An insurance risk model
	7.7 A repair problem
	7.8 Exercising a stock option
	7.9 Verification of the simulation model
	Exercises
	References
8 Statistical analysis of simulated data
	Introduction
	8.1 The sample mean and sample variance
		A method for determining when to stop generating new data
	8.2 Interval estimates of a population mean
	8.3 The bootstrapping technique for estimating mean square errors
	Exercises
	References
9 Variance reduction techniques
	Introduction
	9.1 The use of antithetic variables
	9.2 The use of control variates
	9.3 Variance reduction by conditioning
		9.3.1 Estimating the expected number of renewals by time t
	9.4 Stratified sampling
	9.5 Applications of stratified sampling
		9.5.1 Analyzing systems having Poisson arrivals
		9.5.2 Computing multidimensional integrals of monotone functions
		9.5.3 Compound random vectors
		9.5.4 The use of post-stratification
	9.6 Importance sampling
	9.7 Using common random numbers
	9.8 Evaluating an exotic option
	9.9 Appendix: Verification of antithetic variable approach when estimating the expected value of monotone functions
	Exercises
	References
10 Additional variance reduction techniques
	Introduction
	10.1 The conditional Bernoulli sampling method
	10.2 A simulation estimator based on an identity of Chen–Stein
		10.2.1 When X1, …, Xn are independent
		10.2.2 When X1, …, Xn are dependent
		10.2.3 A post-simulation estimator
	10.3 Using random hazards
	10.4 Normalized importance sampling
	10.5 Latin hypercube sampling
	Exercises
11 Statistical validation techniques
	Introduction
	11.1 Goodness of fit tests
		The chi-square goodness of fit test for discrete data
		The Kolmogorov–Smirnov test for continuous data
	11.2 Goodness of fit tests when some parameters are unspecified
		The discrete data case
		The continuous data case
	11.3 The two-sample problem
	11.4 Validating the assumption of a nonhomogeneous Poisson process
	Exercises
	References
12 Markov chain Monte Carlo methods
	Introduction
	12.1 Markov chains
	12.2 The Hastings–Metropolis algorithm
	12.3 The Gibbs sampler
	12.4 Continuous time Markov chains and a queueing loss model
	12.5 Simulated annealing
	12.6 The sampling importance resampling algorithm
	12.7 Coupling from the past
	Exercises
	References
Index
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