Discrete q-Distributions (eBook)

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 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

"The book is interesting, and ought to be useful for all researchers in the developing area of discrete distribution theory. It is also an ideal and comprehensive textbook for graduate level courses in distribution theory and enumerative combinatorics" Malvina George Vamvakari, Mathematical Reviews, Sept 2017