E-Book
88,99 €

Discrete q-Distributions E-Book

Charalambos A. Charalambides

0,0

88,99 €

oder

Leseprobe lesen

Sammeln Sie Punkte in unserem Gutscheinprogramm und kaufen Sie E-Books und Hörbücher mit bis zu 100% Rabatt.

Mehr erfahren.

Herausgeber: John Wiley & Sons
Kategorie: Wissenschaft und neue Technologien
Sprache: Englisch

Beschreibung

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.

Details

Sie lesen das E-Book in den Legimi-Apps auf:

Android

iOS

von Legimi
zertifizierten E-Readern

Seitenzahl: 282

Veröffentlichungsjahr: 2016

Bewertungen

0,0

Rezensionen(0 Rezensionen)

Leseprobe

Cover

Title Page

Dedication

Preface

About the Author

Chapter 1: Basic q-Combinatorics and q-Hypergeometric Series

1.1 Introduction

1.2

-Factorials and

-Binomial Coefficients

1.3

-Vandermonde's and

-Cauchy's Formulae

1.4

-Binomial and Negative

-Binomial Formulae

1.5 General

-Binomial Formula and

-Exponential Functions

1.6

-Stirling Numbers

1.7 Generalized

-Factorial Coefficients

1.8

-Factorial and

-Binomial Moments

1.9 Reference Notes

1.10 Exercises

Chapter 2: Success probability varying with the number of trials

2.1

-Binomial Distribution of the First Kind

2.2 Negative

-Binomial Distribution of the First Kind

2.3 Heine Distribution

2.4 Heine Stochastic Process

2.5

-Stirling Distributions of the First Kind

2.6 Reference Notes

2.7 Exercises

Chapter 3: Success probability varying with the number of successes

3.1 Negative

-Binomial Distribution of the Second Kind

3.2

-Binomial Distribution of the Second Kind

3.3 Euler Distribution

3.4 Euler Stochastic Process

3.5

-Logarithmic Distribution

3.6

-Stirling Distributions of the Second Kind

3.7 Reference Notes

3.8 Exercises

Chapter 4: Success Probability Varying with the Number of Trials and the Number of Successes

4.1

-Pólya Distribution

4.2

-Hypergeometric Distributions

4.3 Inverse

-Pólya Distribution

4.4 Inverse

-Hypergeometric Distributions

4.5 Generalized

-Factorial Coefficient Distributions

4.6 Reference Notes

4.7 Exercises

Chapter 5: Limiting Distributions

5.1 Introduction

5.2 Stochastic and in Distribution Convergence

5.3 Laws of Large Numbers

5.4 Central Limit Theorems

5.5 Stieltjes–Wigert Distribution as Limiting Distribution

5.6 Reference Notes

5.7 Exercises

Appendix: Hints and Answers to Exercises

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

References

Index

End User License Agreement

Pages

xii

xiii

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

241

242

243

244

245

Guide

Cover

Table of Contents

Preface

Begin Reading

DISCRETE q-DISTRIBUTIONS

CHARALAMBOS A. CHARALAMBIDES

Department of Mathematics

University of Athens, Athens, Greece

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Names: Charalambides, Ch. A., author.

Title: Discrete q-distributions / Charalambos A. Charalambides.

Description: Hoboken, New Jersey : John Wiley & Sons, 2016. | Includes bibliographical references and index.

Identifiers: LCCN 2015031840 (print) | LCCN 2015039486 (ebook) | ISBN 9781119119043 (cloth) | ISBN 9781119119050 (pdf) | ISBN 9781119119104 (epub)

Subjects: LCSH: Distribution (Probability theory) | Stochastic sequences. | Discrete geometry. | Combinatorial geometry.

Classification: LCC QA273.6 .C453 2016 (print) | LCC QA273.6 (ebook) | DDC 519.2/4–dc23

LC record available at http://lccn.loc.gov/2015031840

To the memory of my parentsAngelos and Elpida

Preface

The classical binomial and negative binomial (or Pascal) distributions are defined in the stochastic model of a sequence of independent and identically distributed Bernoulli trials. The Poisson distribution may be considered as a limiting case of the binomial distribution as the number of trials tends to infinity. Also, the logarithmic distribution may be considered as a limiting case of the zero-truncated negative binomial distribution as the number of successes tends to zero.

Poisson (1837) generalized the binomial distribution (and implicitly the negative binomial distribution) by assuming that the probability of success at a trial varies with the number of previous trials. The probability function of the number of successes up to a given number of trials was derived by Platonov (1976) in terms of the generalized signless Stirling numbers of the first kind. Balakrishnan and Nevzorov (1997) obtained this distribution as the distribution of the number of records up to a given time in a general record model.

The negative binomial distribution (and implicitly the binomial distribution) can be generalized to a different direction by assuming that the probability of success at a trial varies with the number of successes occurring in the previous trials. The probability function of the number of successes up to a given number of trials was derived by Woodbury (1949) essentially in terms of the generalized Stirling numbers of the second kind. Sen and Balakrishnan (1999) obtained the distribution of the number of trials up to a given number of successes in connection with a reliability model; their expression was also essentially in terms of the generalized Stirling numbers of the second kind.

It should be noticed that a stochastic model of a sequence of independent Bernoulli trials, in which the probability of success at a trial is assumed to vary with the number of trials and/or the number of successes, is advantageous in the sense that it permits incorporating the experience gained from previous trials and/or successes. If the probability of success at a trial is a very general function of the number of trials and/or the number successes, very little can be inferred from it about the distributions of the various random variables that may be defined in this model. The assumption that the probability of success (or failure) at a trial varies geometrically, with rate (proportion) , leads to the introduction of discrete -distributions. The study of these distributions is greatly facilitated by the wealth of existing -sequences and -functions, in -combinatorics, and the theory of -hypergeometric series.

This book is devoted to the study of discrete -distributions. As to its contents, the following remarks and stressing may be useful. The mathematics prerequisites are modest. They are offered by a basic course in infinitesimal calculus. This should include some power series. The necessary basic -combinatorics and-hypergeometric series are included in an introductory chapter making the entire text self-contained.

In Chapter 1, after introducing the notions of a -power, a -factorial, and a -binomial coefficient of a real number, two -Vandermonde's (-factorial convolution) formulae are derived. Furthermore, two -Cauchy's (-binomial convolution) formulae are presented as a corollary of the -Vandermonde's formulae. Also, the -binomial (-Newton's binomial) and the negative -binomial formulae are obtained. In addition, a general -binomial formula is derived and, as limiting forms of it, -exponential and -logarithmic functions are deduced. The -Stirling numbers of the first and second kind, which are the coefficients of the expansions of -factorials into -powers and of -powers into -factorials, respectively, are presented. Also, the generalized -factorial coefficients are briefly discussed. Moreover, the -factorial and -binomial moments of a discrete -distribution are briefly presented. In addition, two equivalent formulae connecting the usual factorial and binomial moments with the -factorial and -binomial moments, respectively, are deduced. Consequently, the -factorial and -binomial moments, apart from the interest in their own, can be used in the calculation of the usual factorial and binomial moments of a discrete -distribution.

Chapter 2 deals with discrete -distributions defined in the stochastic model of a sequence of independent Bernoulli trials, with success probability at a trial varying geometrically with the number of previous trials. Specifically, assuming that the odds of success at a trial is a geometrically varying (increasing or decreasing) sequence, a -binomial distribution of the first kind and a negative -binomial distribution of the first kind are introduced and studied. In addition, the Heine distribution, which is a -Poisson distribution of the first kind, is obtained as a limiting distribution of the -binomial distribution (or the negative -binomial distribution) of the first kind, as the number of trials (or the number of successes) tends to infinity. Furthermore, considering a stochastic model that is developing in time or space, in which events (successes) may occur at continuous points, a Heine stochastic process, which is a -analogue of a Poisson process, is presented. Also, assuming that the probability of success at a trial is a geometrically varying (increasing or decreasing) sequence, a -Stirling distribution of the first kind is defined and discussed. Finally, supposing that the odds of failure at a trial is a geometrically increasing sequence, another -Stirling distribution of the first kind is obtained.

Chapter 3 is devoted to the study of discrete -distributions defined in the stochastic model of a sequence of independent Bernoulli trials with success probability varying geometrically with the number of previous successes. Introducing the notion of a geometric sequence of (Bernoulli) trials as a sequence of independent Bernoulli trials, with constant probability of success, which is terminated with the occurrence of the first success, an equivalent stochastic model is constructed as follows. A sequence of independent geometric sequences of trials with success probability at a geometric sequence of trials varying (increasing or decreasing)geometrically with the number of previous sequences (successes), is considered. In this model, a negative -binomial distribution of the second kind and a -binomial distribution of the second kind are introduced and examined. In addition, the Euler distribution, which is a -Poisson distribution of the second kind, is obtained as a limiting distribution of the -binomial distribution (or the negative -binomial distribution) of the second kind, as the number of trials (or the number of successes) tends to infinity. Furthermore, considering a stochastic model that is developing in time or space, in which events (successes) may occur at continuous points, an Euler stochastic process, which is a -analogue of a Poisson process, is presented. Also, the -logarithmic distribution is deduced as an approximation of a zero-truncated negative -binomial distribution of the second kind, as the number of successes increases indefinitely. Finally, assuming that the probability of success at a geometric sequence of trials is a geometrically varying (increasing or decreasing) sequence, a -Stirling distribution of the second kind is introduced and discussed.

In Chapter 4, a stochastic model of a sequence of independent Bernoulli trials, with success probability varying geometrically both with the number of trials and the number of successes in a specific manner, is considered. In the first part of this chapter, after introducing a -Pólya urn model, a -Pólya distribution is defined and studied. An approximation of the -Pólya distribution by a -binomial distribution of the second kind is obtained. As a particular case, a -hypergeometric distribution is presented. Also, an inverse -Pólya distribution is introduced and discussed. An approximation of the inverse -Pólya distribution by a negative -binomial distribution of the second kind is obtained. As a particular case, an inverse -hypergeometric distribution is examined. The second part of this chapter is concerned with the particular case in which the probability of success at a trial is a product of a function of the number of previous trials only and another function of the number previous successes and varies geometrically. In this stochastic model, generalized -factorial coefficient distributions are defined and discussed.

In Chapter 5, after introducing the mode of stochastic convergence and the mode of convergence in distribution, the Chebyshev's law of large numbers is presented. In the particular case of a sequence of independent Bernoulli trials, with the probability of success varying with the number of trials, the Poisson's law of large numbers is concluded. In the other particular case of a sequence of independent geometric sequences of (Bernoulli) trials, with the probability of success varying with the number of geometric sequences of trials, another particular case of Chebyshev's law of large numbers is deduced. The central limit theorems for independent and not necessarily identically distributed random variables are presented next. Specifically, the Lyapunov and the Lindeberg–Feller central limit theorems are given and their use in investigating the limiting -distributions are discussed. This chapter is concluded with some local limit theorems, which examine the converges of the probability (mass) functions of particular discrete -deformed distributions to the density function of a Stieltjes–Wigertdistribution.

About the Author

Charalambos A. Charalambidesis professor emeritus of mathematical statistics at the University of Athens, Greece. Dr. Charalambides received a diploma in mathematics (1969) and a Ph.D. in mathematical statistics (1972) from the University of Athens. He was visiting assistant professor at McGill University, Montreal, Canada (1973–1974), visiting associate professor at Temple University, Philadelphia, USA (1985–1986), and visiting professor at the University of Cyprus, Nicosia, Cyprus (1995–1996, 2003–2004, 2007–2008, 2010–2011). Since 1979, he has been an elected member of the International Statistical Institute (ISI). Professor Charalambides' research interests include enumerative combinatorics, combinatorial probability, and parametric inference/point estimation. He is the author of the textbooks Enumerative Combinatorics, Chapman & Hall/CRC Press, 2002, and Combinatorial Methods in Discrete Distributions, John Wiley & Sons, 2005, and co-editor of the volume Probability and Statistical Models with Applications, Chapman & Hall/CRC Press, 2001.

Chapter 1Basic q-Combinatorics and q-Hypergeometric Series

1.1 Introduction

The basic -sequences and -functions of the calculus of -hypergeometric series, which facilitate the study of discrete -distributions, are thoroughly presented in this chapter. More precisely, after introducing the notions of a -power, a -factorial, and a -binomial coefficient of a real number, two -Vandermonde's (-factorial convolution) formulae are derived. Also, two -Cauchy's (-binomial convolution) formulae are presented as a corollary of the two -Vandermonde's formulae. Furthermore, the -binomial and the negative -binomial formulae are obtained. In addition, a general -binomial formula is derived and, as limiting forms of it, -exponential and -logarithmic functions are deduced. The -Stirling numbers of the first and second kind, which are the coefficients of the expansions of -factorials into -powers and of -powers into -factorials, respectively, are presented. Also, the generalized -factorial coefficients are briefly discussed. Moreover, the -factorial and -binomial moments, which, apart from the interest in their own, are used as an intermediate step in the calculation of the usual factorial and binomial moments of a discrete -distribution, are briefly presented. Finally, the probability function of a nonnegative integer-valued random variable is expressed in terms of its -binomial (or -factorial) moments.