UNLV Retrospective Theses & Dissertations 1-1-1996 Topics in combinatorics: A statistical application Susan Diane McCuistion University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/rtds Repository Citation McCuistion, Susan Diane, "Topics in combinatorics: A statistical application" (1996). UNLV Retrospective Theses & Dissertations. 3208. http://dx.doi.org/10.25669/lskx-zpr6 This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. 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TOPICS IN COMBINATORICS: A STATISTICAL APPLICATION by Susan Diane McCuistion A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Department of Mathematical Sciences University of Nevada, Las Vegas August 1996 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 1381031 UMI Microform 1381031 Copyright 1996, 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. The Thesis of Susan Diane McCuistion for the degree of Master of Science in Mathematics is approved. ---------- ChaiTMrson, Peter Shiue, PhJD. Examining Committee Member, Chih-Hsiang Ho, PhTD. Examining Committee Member, Malwane Ananda, Ph.D. Graduate Faculty Representatinve, Laxmi P. Gewali, Ph.D. Dean ofihe Graduate College, Ronald W. Smith, Ph.D. University of Nevada, Las Vegas August, 1996 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT The discrete math area known as combinatorics has specific applications in statistics. In this paper, the relationship between combinatorics and statistics will be explored. Major areas of concentration will include: generating functions, special combinatorial numbers and their use in statistics, and the specific relationship of these functions and numbers to some of the more common discrete distributions found in statistics. Ill Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ABSTRACT.................................................................................................................................... i CHAPTER 1 INTRODUCTION.................................................................................................. 1 CHAPTER 2 GENERATING FUNCTIONS............................................................................ 3 Ordinary Generating Functions ...................................................................................... 4 Exponential Generating Functions .................................................................................5 Probability Generating Functions ................................................................................... 6 Moment Generating Functions ....................................................................................... 8 Cumulants .........................................................................................................................13 CHAPTER 3 COMBINATORIAL NUMBERS RELATED TO STATISTICS................ 16 Stirling Numbers ..............................................................................................................16 Stirling Numbers of the First Kind ..................................................................16 Stirling Numbers of the Second Kind ..............................................................18 Relationships Between Stirling Numbers ...................................................... 22 Bell Numbers...................................................................................................................23 Bell Polynomials ............................................................................................................. 24 Lah Numbers....................................................................................................................26 C-Numbers....................................................................................................................... 29 Combinatorial Applications in Discrete Statistical distributions .............................32 Poisson Distribution ...........................................................................................32 Binomial Distribution ....................................................................................... 35 REFERENCES.............................................................................................................................37 IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I would like to thank my committee members for their time and assistance. In particular, thank you to Dr. Shiue for his direction; Dr. Ho for his words of wisdom, both in and out of class; and Dr. Ananda for being available in my time of need. I would also like to thank Helena Murvosh and Donna Frasier for their word processing expertise and help in completing this paper. On a personal level, thank you to my mother and father for supporting me and my decisions. Thank you to my brother, “the word man”, for helping me through creative blocks. I would also like to thank my friends, particularly Brad Schultz, Margie Wells and Nancy Millet, for their kind words and understanding during stressful times. Finally, but most importantly, thank you to my husband, Danny, for his endless love and boundless support Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 INTRODUCTION Of the two main divisions of mathematics, continuous and discrete, most attention is paid in basic courses to the continuous. Integral calculus is emphasized more than the calculus of finite differences. In statistics, a short amount of time is spent on discrete variables and distributions before they become an afterthought to any theorems stated for continuous variables. The discrete mathematics area known as combinatorics has a useful tool for building a bridge between the discrete and continuous areas of mathematics, specifically, the Stirling numbers. Jordan notes ""Stirling’s Numbers are of the greatest utility in Mathematical Calculus. This however has not been fully recognised [sic]; the numbers have been neglected, and are seldom used” [11], (p. 143). The numbers have popped up from time to time, with different names and notations, and perhaps the different authors themselves did not realize they dealt with the same numbers. Jordan continues, ""Stirling’s numbers are as important or even more so than Bernoulli’s numbers; they should be placed in the centre [sic] of the Calculus of Finite Differences” [11], (p. 143). This paper explores the relationship of certain areas of combinatorics to statistics. In particular, the second chapter deals with generating functions, one of the foundations Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 of combinatorial analysis. Riordan states, “even in the