Discrete q–Distributions

A self–contained study of the various applications and developments of discrete distribution theory

Written by a well–known researcher in the field, Discrete q–Distributions features an organized presentation of discrete q–distributions defined on the stochastic model of a sequence of independent Bernoulli trials.  In an effort to keep the book self–contained, the author covers all of the necessary basic q–sequences and q–functions.

The book begins with an introduction of the notions of a q–power, a q–factorial, and a q–binomial coefficient and proceeds to discuss the basic q–Combinatorics and q–hypergeometric series. Next, the book addresses discrete q–distributions 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 q–Distributions also features:
Discussions of the definitions and theorems that highlight key concepts and results Several worked out 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 less experienced readers gain a better understanding of the content An up–to–date bibliography that includes the latest trends and advances in the field and is a collective source for further research 

A unique reference for researchers and practitioners in statistics, mathematics, physics, engineering, and other applied sciences, Discrete q–Distributions is also an appropriate textbook for graduate–level courses in discrete statistical distributions, distribution theory, and combinatorics.

Charalambos A. Charalambides, PhD, is Professor Emeritus in the Department of Mathematics at the University of Athens, Greece. An elected member of the International Statistical Institute, his research interests include enumerative combinatorics, combinatorial probability, and parametric inference/point estimation. He is the author of Combinatorial Methods of Discrete Distributions, also published by Wiley .

Preface ix

1 Basic q–combinatorics and q–hypergeometric series 1

1.1 Introduction 1

1.2 q–Factorials and q–binomial coefficients 1

1.3 q–Vandermonde s and q–Cauchy s formulae 10

1.4 q–Binomial and negative q–binomial formulae 16

1.5 General q–binomial formula and q–exponential functions 24

1.6 q–Stirling numbers 26

1.7 Generalized q–factorial coefficients 36

1.8 q–Factorial and q–binomial moments 42

1.9 Reference notes 44

1.10 Exercises 45

2 Success probability varying with the number of trials 61

2.1 q–Binomial distribution of the first kind 61

2.2 Negative q–binomial distribution of the first kind 66

2.3 Heine distribution 69

2.4 Heine stochastic process 73

2.5 q–Stirling distributions of the first kind 77

2.6 Reference notes 84

2.7 Exercises 85

3 Success probability varying with the number of successes 95

3.1 Negative q–binomial distribution of the second kind 95

3.2 q–Binomial distribution of the second kind 99

3.3 Euler distribution 103

3.4 Euler stochastic process 107

3.5 q–Logarithmic distribution 111

3.6 q–Stirling distributions of the second kind 114

3.7 Reference notes 119

3.8 Exercises 121

4 Success probability varying with the number of successes and the number of trials 131

4.1 q–Pólya distribution 131

4.2 q–Hypergeometric distributions 139

4.3 Inverse q–Pólya distribution 145

4.4 Inverse q–hypergeometric distributions 149

4.5 Generalized q–factorial coefficient distributions 150

4.6 Reference notes 159

4.7 Exercises 160

5 Limiting distributions 167

5.1 Introduction 167

5.2 Stochastic and in distribution convergence 168

5.3 Laws of large numbers 170

5.4 Central limit theorems 175

5.5 Stieltjes–Wigert distribution as limiting distribution 179

5.6 Reference notes 186

5.7 Exercises 186

Appendix Hints and answers to exercises 191

Bibliography 229

Index 233