Discrete qDistributionsISBN: 9781119119043
264 pages
March 2016

Description
A selfcontained study of the various applications and developments of discrete distribution theory
Written by a wellknown researcher in the field, Discrete qDistributions features an organized presentation of discrete qdistributions based on the stochastic model of a sequence of independent Bernoulli trials. In an effort to keep the book selfcontained, the author covers all of the necessary basic qsequences and qfunctions.
The book begins with an introduction of the notions of a qpower, a qfactorial, and a qbinomial coefficient and proceeds to discuss the basic qcombinatorics and qhypergeometric series. Next, the book addresses discrete qdistributions with success probability at a trial varying geometrically, with rate q, either with the number of previous trials or with the number of previous successes. Further, the book examines two interesting stochastic models with success probability at any trial varying geometrically both with the number of trials and the number of successes and presents local and global limit theorems. Discrete qDistributions also features:
 Discussions of the definitions and theorems that highlight key concepts and results
 Several worked examples that illustrate the applications of the presented theory
 Numerous exercises at varying levels of difficulty that consolidate the concepts and results as well as complement, extend, or generalize the results
 Detailed hints and answers to all the exercises in an appendix to help lessexperienced readers gain a better understanding of the content
 An uptodate bibliography that includes the latest trends and advances in the field and provides a collective source for further research
 An Instructor’s Solutions Manual available on a companion website
A unique reference for researchers and practitioners in statistics, mathematics, physics, engineering, and other applied sciences, Discrete qDistributions is also an appropriate textbook for graduatelevel courses in discrete statistical distributions, distribution theory, and combinatorics.
Table of Contents
Preface ix
1 Basicqcombinatorics and qhypergeometric series 1
1.1 Introduction 1
1.2 qFactorials and qbinomial coefficients 2
1.3 qVandermonde’s and qCauchy’s formulae 10
1.4 qBinomial and negative qbinomial formulae 16
1.5 General qbinomial formula and qexponential functions 24
1.6 qStirling numbers 26
1.7 Generalized qfactorial coefficients 36
1.8 qFactorial and qbinomial moments 42
1.9 Reference notes 45
1.10 Exercises 46
2 Success probability varying with the number of trials 61
2.1 qbinomial distribution of the first kind 61
2.2 Negative qbinomial distribution of the first kind 66
2.3 Heine distribution 69
2.4 Heine stochastic process 73
2.5 qStirling distributions of the first kind 77
2.6 Reference notes 85
2.7 Exercises 86
3 Success probability varying with the number of successes 97
3.1 Negative qbinomial distribution of the second kind 97
3.2 qBinomial distribution of the second kind 102
3.3 Euler distribution 105
3.4 Euler stochastic process 109
3.5 qLogarithmic distribution 114
3.6 qStirling distributions of the second kind 117
3.7 Reference notes 122
3.8 Exercises 123
4 Success probability varying with the number of successes and the number of trials 135
4.1 qPólya distribution 135
4.2 qHypergeometric distributions 144
4.3 Inverse qpólya distribution 150
4.4 Inverse qhypergeometric distributions 154
4.5 Generalized qfactorial coefficient distributions 155
4.6 Reference notes 164
4.7 Exercises 165
5 Limiting distributions 173
5.1 Introduction 173
5.2 Stochastic and in distribution convergence 174
5.3 Laws of large numbers 176
5.4 Central limit theorems 181
5.5 Stieltjes–wigert distribution as limiting distribution 185
5.6 Reference notes 193
5.7 Exercises 193
Appendix Hints and answers to exercises 197
References 235
Index 241