Computational Probability (eBook)

Front Cover 1
Computational Probability 4
Copyright Page 5
Table of Contents 8
Contributors 10
Preface 12
Chapter 1. Some Ideas in Computational Probability 14
References 23
Chapter 2. Computational Problems Related to the Galton— Watson Press 24
I. THE PROBABILITY DISTRIBUTIONS OF PHASE TYPE 25
2. THE SUCCESSIVE GENERATION SIZES IN A GALTON-WATSON PROCESS 33
3. A GALTON-WATSON PROCESS EMBEDDED IN 
42 
4. THE MACIMUM GENERATION SIZE BEFORE EXTINCTION 46
5. THE PROBABILITY OF EVENTUAL ABSORPTION 49
REFERENCES 50
Chapter 3. Central Limit Analogues for Markov Population Processes 52
1. SUMMARY 53
2. A CENTRAL LIMIT THEOREM FOR UNIVARIATE PROCESSES 54
3. MULTIVARIATE MARKOV PROCESSES 61
4. PROBLEM OF THE TWO SEXES 62
5. MULTIVARIATE GENERALIZED EHRENFEST PROCESS 68
6. GROWTH OF TWO CITIES 72
7. SOME RELATED RESULTS 74
REFERENCES 81
Chapter 4. The Stochastic Nature of Pension Costs 84
SUMMARY 84
INTRODUCTION 84
THE PROBABILITY OF A GIVEN NUMBER OF EMPLOYEES AT EACH AGE 86
THE PROBABILITY THAT EXPECTED RETIREMENT COSTS EXCEED ACTUAL RETIREMENT COSTS 90
WORKING FORMULAE FOR THE DETERMINATION OF THE PROBABLE ADEQUACY OF EXPECTED PENSION COSTS 93
ESTIMATES OF THE ADEQUACY OF EXPECTED PENSION COSTS 93
COMMENT 99
REFERENCES 101
Chapter 5. An Approximation Method to Calculate the Value of a Maturity Guarantee under a Level-Premium Equity-Based Contract 104
INTRODUCTION AND SIMULATION METHOD 104
THE APPROXIMATION METHOD 107
ACKNOWLEDGEMENTS 111
REFERENCES 112
Chapter 6. Efficient Sorting by Computer: An Introduction 114
SUMMARY 114
INTRODUCTION 115
INTERNAL SORTING: HEAPSORT 116
EXTERNAL SORTING: REPLACEMENT SELECTION AND POLYPHASE MERGE 130
CONCLUSION 142
REFERENCES 143
Chapter 7. Symbolic Information Pressing 144
I. ANALYSIS OF ALGORITHMS 144
II. AN ARTIFICIAL INTELLIGENCE APPROACH TO BUSINESS DATA PROCESSING 155
APPENDIX: GRAPHS AND TREES 165
REFERENCES 166
Chapter 8. APL for Actuaries 168
CONCLUDING COMMENTS 183
Chapter 9. Backward Population Projection by a Generalized Inverse 186
1. THE PROBLEM 187
2. A GENERALIZED INVERSE OF THE LESLIE MATRIX 188
3. RELATION TO THE CLASSICAL INVERSE 192
4. APPLICATION 194
REFERENCES 196
Chapter 10. Accounting Principles for Life Insurance: Reflections on Language and Notation 198
INTRODUCTION 198
PART 1. ACCOUNTING PRINCIPLES 200
REFERENCES 204
Chapter 11. Reversionary Annuities as Applied to the Evaluation of Law Amendment Factors 244
INTRODUCTION 244
I. THE LOGISTICS UNDERLYING ANNUITY ASSIGNMENTS 245
II. THE IMPLEMENTATION FOR COMPUTATION 247
III. AN ILLUSTRATION. FECA PROPOSED BENEFIT LEVEL CHANGE FOR FATAL SURVIVOR TO MEET THE NATIONAL COMMISSION ON STATE WORKMEN'S COMPENSATION LAWS LEVEL 250
REFERENCES 260
Chapter 12. Nonlife Business and Inflation 262
Chapter 13. Numerical Fourier Inversion 270
Chapter 14. Some Practical Considerations in Connection with the Calculation of Stop-Loss Premiums 274
1. WHY THE EXPONENTIAL PRINCIPLE 274
2. MEREU'S FORMULA, OR WHY THE TAIL IS NOT A PROBLEM 276
3. THE COMPOUND POISSON DISTRIBUTION 278
4. THE METHOD OF DISPERSAL 280
5. THE METHOD OF TRUNCATION 281
6. AN ILLUSTRATIVE EXAMPLE 282
REFERENCES 299
Chapter 15. Simulation of a Multirisk Collective Model 300
1. INTRODUCTION 300
2. SIMULATION OF THE MODEL 304
3. IMPLEMENTATION OF THE ALGORITHM 305
4. EXAMPLES 307
5. DISCUSSION OF OTHER REFERENCES 313
REFERENCES 313
Chapter 16. Experimental Computation 316
1. INTRODUCTION 316
2. INSTRUCTION 318
3. RESEARCH 320
REFERENCES 323
Chapter 17. Partitioning for Homogeneity 324
SUMMARY 324
INTRODUCTION 324
STATISTICAL APPLICATIONS 326
OPTIMAL PARTITIONS 329
REFERENCES 336
Chapter 18. Correlates of Life Insurance Lapsation 338
Chapter 19. Some Practical Notes about Solving the Lundberg Risk Equation 350
REFERENCES 353