# Contents

1. Introduction

• Available Software
• What This Book Does Not Contain
• Conventions

2. Estimating Volume and Count

• Volume
• Error and Sample Size Considerations
• Confidence Intervals
• Exploiting Regional Bounds
• Relative Error
• Network Reliability
• Multivariable Integration
• Exploiting Function Bounds
• Exploiting ParameterBounds
• Restricting the Sampling Region
• Reducing the Sampling Dimension
• Counting Problems
• Sensitivity Analysis
• Simultaneous Confidence Intervals
• Ratio Estimation
• Sequential Estimation

3. Generating Samples

• Independence and Dependence
• Inverse Transform Method
• Cutpoint Method
• Composition Method
• Alias Method
• Acceptance-Rejection Method
• Ratio-of-Uniforms Method
• Exact-Approximation Method
• Algorithms for Selected Distributions
• Exponential Distribution
• Normal Distribution
• Lognormal Distribution
• Cauchy Distribution
• Gamma Distribution
• Beta Distribution
• Student’s t Distribution
• Snedecor’s F Distribution
• Revisiting the Ratio-of-Uniforms Method
• Poisson Distribution
• Binomial Distribution
• Hypergeometric Distribution
• Geometric Distribution
• Negative Binomial Distribution
• Multivariate Normal Distribution
• Multinomial Distribution
• Order Statistics
• Sampling Without Replacement and Permutations
• Points in and on a Simplex
• Points in and on a Hyperellipsoid
• Bernoulli Trials
• Sampling from a Changing Probability Table
• Random Spanning Trees

4. Increasing Efficiency

• Importance Sampling
• Control Variates
• Stratified Sampling
• Inducing Correlation
• Conditonal Monte Carlo

5. Random Tours

• Markov Processes
• Random Walk
• Markov Time
• Score Processes
• Neutron Transport
• Buffer Exceedance on a Production Line
• Fredholm Equations of the Second Kind
• Catastrophic Failure
• First Passage Time
• Random StoppingTime
• Generating Random Points from a Target Distribution
• Generating a Uniformly Distributed Point on a Finite Set
• Generating All Coordinates in a Bounded Region on Each Step
• Metropolis Method
• Sampling Coordinates One at a Time
• Markov Random Fields
• Gibbs Sampling
• Simulated Annealing
• Bayesian Posterior Distributions
• Edge Effects
• Time to Stationarity
• Spectral Structure
• Bounds on Error
• Varying Initial Conditions
• Random Walk on a Hypercube
• Conductance
• More About a Random Walk on a Hypercube
• An Alternative Error Bound for Stationarity
• Sampling from a Hyperrectangular Grid
• Sampling from a Hyperrectangle
• Product Estimator
• Estimating the Volume of a Convex Body
• Estimating the Permanent
• Coupling
• Strong Markov Time
• Strong Stationary Dual Process
• Thresholds

6. Designing and Analyzing Sample Paths

• Problem Context
• A First Approach to Computing Confidence Intervals
• Warm-Up Analysis
• Choosing a “Good” Initial State or a “Good” Initial Distribution
• Strictly Stationary Stochastic Processes
• Optimal Choice of Sample Path Length t and Number of Replications n
• Estimating Required Sample Path Length
• Characterizing Convergence
• An Alternative View of the Variance of the Sample Mean
• Batch Means Method
• Batch Means Analysis Programs
• Regenerative Processes
• Selecting an Optimal Acceptance Scheme for Metropolis Sampling

7. Generating Pseudorandom Numbers

• Linear Recurrence Generators
• Prime Modulus Generators
• Power-of-Two Modulus Generators
• Mixed Congruential Generators
• Implementation and Portability
• Apparent Randomness
• Spectral Test
• Minimal Number of Parallel Hyperplanes
• Distance Between Points
• Discrepancy
• Beyer Quotient
• Empirical Assessments
• Combining Linear Congruential Generators
• j-Step Linear Recurrence
• Feedback Shift Register Generators
• Generalized Feedback Shift Register Generators
• Nonlinear Generators