Simulation, Sixth Edition continues to introduce aspiring and practicing actuaries, engineers, computer scientists and others to the practical aspects of constructing computerized simulation studies to analyze and interpret real phenomena. Readers will learn to apply the results of these analyses to problems in a wide variety of fields to obtain effective, accurate solutions and make predictions. By explaining how a computer can be used to generate random numbers and how to use these random numbers to generate the behavior of a stochastic model over time, this book presents the statistics needed to analyze simulated data and validate simulation models.
1. 1. Introduction - p. 1 2. 2. Elements of probability - p. 5 3. Abstract - p. 5 4. 2.1 Sample space and events - p. page5 5. 2.2 Axioms of probability - p. page5 6. 2.3 Conditional probability and independence - p. page6 7. 2.4 Random variables - p. 9 8. 2.5 Expectation - p. 11 9. 2.6 Variance - p. 13 10. 2.7 Chebyshev's inequality and the laws of large numbers - p. 15 11. 2.8 Some discrete random variables - p. page17 12. 2.9 Continuous random variables - p. 23 13. 2.10 Conditional expectation and conditional variance - p. page30 14. Exercises - p. 32 15. References - p. 37 16. 3. Random numbers - p. 39 17. 4. Generating discrete random variables - p. 47 18. 5. Generating continuous random variables - p. 69 19. Abstract - p. 69 20. Introduction - p. page69 21. 5.1 The inverse transform algorithm - p. page69 22. 5.2 The rejection method - p. 73 23. 5.3 The polar method for generating normal random variables - p. 86 24. 5.4 Generating a Poisson process - p. 85 25. 5.5 Generating a nonhomogeneous Poisson process - p. 87 26. 5.6 Simulating a two-dimensional Poisson process - p. 90 27. Exercises - p. 94 28. References - p. 97 29. 6. The multivariate normal distribution and copulas - p. 99 30. 7. The discrete event simulation approach - p. 111 31. 8. Statistical analysis of simulated data - p. 133 32. 9. Variance reduction techniques - p. 151 33. Abstract - p. 151 34. Introduction - p. page151 35. 9.1 The use of antithetic variables - p. 160 36. 9.2 The use of control variates - p. 159 37. 9.3 Variance reduction by conditioning - p. 166 38. 9.4 Stratified sampling - p. page180 39. 9.5 Applications of stratified sampling - p. 190 40. 9.6 Importance sampling - p. 199 41. 9.7 Using common random numbers - p. 212 42. 9.8 Evaluating an exotic option - p. 213 43. 9.9 Appendix: Verification of antithetic variable approach when estimating the expected value of monotone functions - p. 217 44. Exercises - p. 219 45. References - p. 227 46. 10. Additional variance reduction techniques - p. 229 47. Abstract - p. 229 48. Introduction - p. page229 49. 10.1 The conditional Bernoulli sampling method - p. page229 50. 10.2 A simulation estimator based on an identity of Chen–Stein - p. 233 51. 10.3 Using random hazards - p. 241 52. 10.4 Normalized importance sampling - p. 246 53. 10.5 Latin hypercube sampling - p. 250 54. Exercises - p. 252 55. 11. Statistical validation techniques - p. 255 56. 12. Markov chain Monte Carlo methods - p. 279 57. Abstract - p. 279 58. Introduction - p. page279 59. 12.1 Markov chains - p. page279 60. 12.2 The Hastings–Metropolis algorithm - p. 282 61. 12.3 The Gibbs sampler - p. 284 62. 12.4 Continuous time Markov chains and a queueing loss model - p. 294 63. 12.5 Simulated annealing - p. 298 64. 12.6 The sampling importance resampling algorithm - p. 300 65. 12.7 Coupling from the past - p. 304 66. Exercises - p. 306 67. References - p. 308 68. Index - p. 311
Simulation, Sixth Edition continues to introduce aspiring and practicing actuaries, engineers, computer scientists and others to the practical aspects of constructing computerized simulation studies to analyze and interpret real phenomena. Readers will learn to apply the results of these analyses to problems in a wide variety of fields to obtain effective, accurate solutions and make predictions. By explaining how a computer can be used to generate random numbers and how to use these random numbers to generate the behavior of a stochastic model over time, this book presents the statistics needed to analyze simulated data and validate simulation models.
1. 1. Introduction - p. 1 2. 2. Elements of probability - p. 5 3. Abstract - p. 5 4. 2.1 Sample space and events - p. page5 5. 2.2 Axioms of probability - p. page5 6. 2.3 Conditional probability and independence - p. page6 7. 2.4 Random variables - p. 9 8. 2.5 Expectation - p. 11 9. 2.6 Variance - p. 13 10. 2.7 Chebyshev's inequality and the laws of large numbers - p. 15 11. 2.8 Some discrete random variables - p. page17 12. 2.9 Continuous random variables - p. 23 13. 2.10 Conditional expectation and conditional variance - p. page30 14. Exercises - p. 32 15. References - p. 37 16. 3. Random numbers - p. 39 17. 4. Generating discrete random variables - p. 47 18. 5. Generating continuous random variables - p. 69 19. Abstract - p. 69 20. Introduction - p. page69 21. 5.1 The inverse transform algorithm - p. page69 22. 5.2 The rejection method - p. 73 23. 5.3 The polar method for generating normal random variables - p. 86 24. 5.4 Generating a Poisson process - p. 85 25. 5.5 Generating a nonhomogeneous Poisson process - p. 87 26. 5.6 Simulating a two-dimensional Poisson process - p. 90 27. Exercises - p. 94 28. References - p. 97 29. 6. The multivariate normal distribution and copulas - p. 99 30. 7. The discrete event simulation approach - p. 111 31. 8. Statistical analysis of simulated data - p. 133 32. 9. Variance reduction techniques - p. 151 33. Abstract - p. 151 34. Introduction - p. page151 35. 9.1 The use of antithetic variables - p. 160 36. 9.2 The use of control variates - p. 159 37. 9.3 Variance reduction by conditioning - p. 166 38. 9.4 Stratified sampling - p. page180 39. 9.5 Applications of stratified sampling - p. 190 40. 9.6 Importance sampling - p. 199 41. 9.7 Using common random numbers - p. 212 42. 9.8 Evaluating an exotic option - p. 213 43. 9.9 Appendix: Verification of antithetic variable approach when estimating the expected value of monotone functions - p. 217 44. Exercises - p. 219 45. References - p. 227 46. 10. Additional variance reduction techniques - p. 229 47. Abstract - p. 229 48. Introduction - p. page229 49. 10.1 The conditional Bernoulli sampling method - p. page229 50. 10.2 A simulation estimator based on an identity of Chen–Stein - p. 233 51. 10.3 Using random hazards - p. 241 52. 10.4 Normalized importance sampling - p. 246 53. 10.5 Latin hypercube sampling - p. 250 54. Exercises - p. 252 55. 11. Statistical validation techniques - p. 255 56. 12. Markov chain Monte Carlo methods - p. 279 57. Abstract - p. 279 58. Introduction - p. page279 59. 12.1 Markov chains - p. page279 60. 12.2 The Hastings–Metropolis algorithm - p. 282 61. 12.3 The Gibbs sampler - p. 284 62. 12.4 Continuous time Markov chains and a queueing loss model - p. 294 63. 12.5 Simulated annealing - p. 298 64. 12.6 The sampling importance resampling algorithm - p. 300 65. 12.7 Coupling from the past - p. 304 66. Exercises - p. 306 67. References - p. 308 68. Index - p. 311
