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Algorithms for Operations on Probability Distributions in a Computer Algebra System W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects 2001 Algorithms for operations on probability distributions in a computer algebra system Diane Lynn Evans College of William & Mary - Arts & Sciences Follow this and additional works at: https://scholarworks.wm.edu/etd Part of the Mathematics Commons, and the Statistics and Probability Commons Recommended Citation Evans, Diane Lynn, "Algorithms for operations on probability distributions in a computer algebra system" (2001). Dissertations, Theses, and Masters Projects. Paper 1539623382. https://dx.doi.org/doi:10.21220/s2-bath-8582 This Dissertation is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. Reproduced with with permission permission of the of copyright the copyright owner. owner.Further reproductionFurther reproduction prohibited without prohibited permission. without permission. ALGORITHMS FOR OPERATIONS ON PROBABILITY DISTRIBUTIONS IN A COMPUTER ALGEBRA SYSTEM A Dissertation Presented to The Faculty of the Department of Applied Science The College of William & Mary in Virginia In Partial Fulfillment Of the Requirements for the Degree of Doctor of Philosophy by Diane Lynn Evans July 2001 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3026405 Copyright 2001 by Evans, Diane Lynn All rights reserved. ___ ® UMI UMI Microform 3026405 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPROVAL SHEET This dissertation is submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Diane L. Evans, Author APPROVED, July 2001 u Lawrence Leemis * * l3Rex Av MKincaid in Am/1 ^ IS<W) Dennis Manos i t ' John Drew Sidney Lawrence Mathematics Department ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents Acknowledgements v List of Tables vi List of Figures vii A bstract x 1 Introduction 2 1.1 Notation and Nomenclature .......................................................................... 1 1 1.2 Introductory E xam ples .................................................................................. 1 2 2 Data Structure 17 2.1 Standard Discrete Data Structure Form ats .............................................. 27 2 .2 The Six Functional Representations ........................................................... 33 2.3 Algorithms for Fundamental Procedures ................................................. 51 3 Order Statistics 56 3.1 Implementation for Discrete Populations ................................................. 58 3.2 E x a m p le s ......................................................................................................... 70 3.3 Range S ta tistic s ............................................................................................... 79 3.4 Eliminating Resampling Error in Bootstrapping ................................... 8 8 4 Convolutions and Products 97 4.1 Conceptual Fram ew ork .................................................................................. 104 4.2 A lg o rith m ......................................................................................................... 117 4.3 Im plem entatio n ............................................................................................... 120 4.4 Examples .......................................................................................................... 123 4.5 Products of Random Variables with Finite Supports ............................. 133 5 Transformations 145 5.1 T h e o ry ............................................................................................................... 146 5.2 Im plem entation .............................................................................................. 149 5.3 A pplications ..................................................................................................... 164 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 Minimums and Maximums 170 6.1 PDF of the Minimum ...................................................................................... 172 6.2 PDF of the Maximum ................................................................................... 185 7 Algorithms for Operations on Continuous Distributions 195 7.1 Existence Conditions for PD Fs .................................................................. 195 7.2 Method of Moments Estimation .................................................................. 200 7.3 Maximum Likelihood Estimation with Right Censoring ....................... 207 7.4 Mixture and Truncate P ro c e d u re s ........................................................... 212 8 Survival Distributions Satisfying Benford’s Law 218 8.1 Benford’s L a w ................................................................................................... 218 8 . 2 Parametric Survival Distributions ............................................................... 220 8.3 Conditions for Conformance to Benford’s L aw ................ 222 8.4 Variate Generation ......................................................................................... 230 8.5 C onclusio ns ...................................................................................................... 232 9 Input Modeling 233 9.1 E x a m p le s ......................................................................................................... 234 9.2 Further work ................................................................................................... 245 10 APPLications 247 10.1 Kolmogorov-Smirnov Test Statistic for Estimated Parameters .... 247 10.2 O th e r s ................................................................................................................. 258 11 Future Work 271 A Algorithm for OrderStat 274 B Maple Code for NextCombination and NextPermutation 277 C Determining Candidate Sums for the Heap 282 D Algorithm for BruteForceMethod 284 E A lgorithm for MovingHeapMethod 285 F APPL Code for Benford 287 Bibliography 288 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgement s I would like to thank: My committee members: Dr. Drew, Dr. Kincaid, Dr. Manos, and Dr. Lawrence for their careful reading and suggestions of my dissertation and for being great instructors, both in and out of the classroom; Dr. Andrew Glen for allowing me to become part of “APPL” and showing me the ropes to becoming a Maple programmer; The Operations Research faculty at The College of William & Mary for out­ standing instruction and a strong probability and statistics foundation; Dr. Frank Carroll for being my mentor, friend, and mathematical “sounding board” for many years; The Clare Boothe Luce foundation for their generous fellowship that allowed me to continue my education and have the freedom to delve into my research; Dr. Larry Leemis for being himself: an excellent teacher, researcher, and advisor. I have spent three of the best years of my life working with him and will always admire and respect him in many ways. The time he has spent with me will always be appreciated, and I hope that someday I may also make such a positive impact, mathematically and otherwise, on another person’s life. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables 1 .1 Observed horse kick fatalities ......................................................................... 15 2.1 Discrete random variable support categories .............................................. 24 2.2 The six functional representations of a random variable X .................... 34 2.3 Distribution representation relationships .................................................... 35 3.1 Categorization of discrete order statistics with associated examples. 71 3.2 Rat survival data............................................................................................... 90 3.3 Bootstrap estimates of the standard error of the median ........................ 91 3.4 Bootstrap estimates of the standard error of the mean ........................... 93 4.1 Comparison of BruteForceMethod and MovingHeapMethod ................... 123 4.2 Probability table for a convolution ............................................................... 130 4.3 The exact probabilities and normal PDF approximations for Pr(S = s) for s = 7 , 8 , ..., 21............................................................................................ 131 5.1 Categories for computing the PDF of the random variable Y = g(X ) when X is a discrete random variable with support Qx hi a Dot support form at .................................................................................................................. 157 5.2 Life tests on a three-component system .....................................................
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