Discrete q-Distributions


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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 based 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 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 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 q-Distributions is also an appropriate textbook for graduate-level courses in discrete statistical distributions, distribution theory, and combinatorics.

Preface ix

1 Basicq-combinatorics and q-hypergeometric series 1

1.1 Introduction 1

1.2 q-Factorials and q-binomial coefficients 2

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 45

1.10 Exercises 46

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 85

2.7 Exercises 86

3 Success probability varying with the number of successes 97

3.1 Negative q-binomial distribution of the second kind 97

3.2 q-Binomial distribution of the second kind 102

3.3 Euler distribution 105

3.4 Euler stochastic process 109

3.5 q-Logarithmic distribution 114

3.6 q-Stirling 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 q-Pólya distribution 135

4.2 q-Hypergeometric distributions 144

4.3 Inverse q-pólya distribution 150

4.4 Inverse q-hypergeometric distributions 154

4.5 Generalized q-factorial 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

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