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A Probability Programming Language: Development and Applications W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects 1998 A probability programming language: Development and applications Andrew Gordon Glen College of William & Mary - Arts & Sciences Follow this and additional works at: https://scholarworks.wm.edu/etd Part of the Computer Sciences Commons, and the Statistics and Probability Commons Recommended Citation Glen, Andrew Gordon, "A probability programming language: Development and applications" (1998). Dissertations, Theses, and Masters Projects. Paper 1539623920. https://dx.doi.org/doi:10.21220/s2-1tqv-w897 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]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely afreet reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with with permission permission of the of copyright the copyright owner. owner.Further reproductionFurther reproduction prohibited without prohibited permission. without permission. A Probability Programming Language: Development and Applications 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 Andrew G. Glen 1998 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9904264 Copyright 19 9 9 by Glen, Andrew Gordon All rights reserved. UMI Microform 9904264 Copyright 1998, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 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 Andrew G. Glen APPROVED. January 1998 Low 'Mad I Lawrence M. Leemis, Dissertation Advisor . J/LS S A n /V/ . .X AS . ^ John H. Drew Sidney H. Lawrence Rex K. Kincaid / \ AA^ Donald R. Barr, Outside Examiner Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C ontents 1 Introduction 2 1.1 General ........................................................................................................................ 2 1.2 Literature review ...................................................................................................... 6 1.3 Outline of the dissertation .................................................................................... 7 1.4 Notation and nomenclature ................................................................................ S 2 Software Development 10 2.1 The common data structure ................................................................................ 13 2.2 Common continuous, univariate distributions ............................................... 16 2.3 The six representations of distributions .......................................................... IS 2.4 VerifyPDF.............................................................................................................. 21 2.5 MedianRV ..................................................................................................................... 24 2.6 DisplayRV ................................................................................................................ 25 2.7 PlotDist .................................................................................................................... 26 2.S Expect at ionRV ......................................................................................................... 28 2.9 Transform .............................................................................................................. 29 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.10 OrderStat ................................................................................................................ 31 2.11 ProductRV and ProductllD ............................................................................ 33 2.12 SumRV and Sum llD .............................................................................................. 36 2.13 MinimumRV................................................................................................................ 3S 2.14 MaximumRV................................................................................................................ 39 2.15 Maximum likelihood e s tim a tio n .......................................................................... 41 3 Transformations of Univariate Random Variables 44 3.1 In tro d u ctio n .................................................................................................................. 44 3.2 T h e o re m ......................................................................................................................... 47 3.3 Im p lem en tatio n .......................................................................................................... 50 3.4 Examples ..................................................................................................................... 53 3.5 C onclusion ..................................................................................................................... 59 4 Products of Random Variables 61 4.1 Introduction ................................................................................................................. 61 4.2 T h e o re m ........................................................................................................................ 62 4.3 Im p lem en tatio n .......................................................................................................... 65 4.4 Examples ..................................................................................................................... 68 4.5 C onclusion ..................................................................................................................... 75 5 Computing the CDF of the Kolmogorov-Smirnov Test Statistic 76 5.1 Introduction ................................................................................................................. 76 5.2 Literature r e v ie w ..................................................................................................... 78 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. •5.3 Computing the distribution of Dn ...................................................................... 79 5.3.1 Phase 1: Partition the support of Dn — ^ ....................................... S2 5.3.2 Phase 2: Define the A m a tr ic e s ............................................................ S3 5.3.3 Phase 3: Set limits on the appropriateintegrals ............................. S9 5.3.4 Phase 4: Shift the distribution................................................................ 96 5.4 Critical values and significance levels .............................................................. 99 5.5 Conclusion ..................................................................................................................... 100 6 Goodness of Fit using Order Statistics 102 6.1 Introduction ................................................................................................................. 102 6.2 The V -vecto v .............................................................................................................. 104 6.3 Improving computation of V ................................................................................ 106 6.4 Goodness-of-fit testing ..........................................................................................
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