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(cid:2) DISCRETE q-DISTRIBUTIONS (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) DISCRETE q-DISTRIBUTIONS CHARALAMBOSA.CHARALAMBIDES DepartmentofMathematics UniversityofAthens,Athens,Greece (cid:2) (cid:2) (cid:2) (cid:2) Copyright©2016byJohnWiley&Sons,Inc.Allrightsreserved PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey PublishedsimultaneouslyinCanada Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformor byanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptas permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeeto theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,(978)750-8400,fax (978)750-4470,oronthewebatwww.copyright.com.RequeststothePublisherforpermissionshould beaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ 07030,(201)748-6011,fax(201)748-6008,oronlineathttp://www.wiley.com/go/permissions. LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsin preparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales representativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbesuitable foryoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthepublishernor authorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnotlimitedto special,incidental,consequential,orotherdamages. Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontactour CustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnitedStatesat (cid:2) (317)572-3993orfax(317)572-4002. (cid:2) Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmay notbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitourwebsiteat www.wiley.com. LibraryofCongressCataloging-in-PublicationData: Names:Charalambides,Ch.A.,author. Title:Discreteq-distributions/CharalambosA.Charalambides. Description:Hoboken,NewJersey:JohnWiley&Sons,2016.|Includes bibliographicalreferencesandindex. Identifiers:LCCN2015031840(print)|LCCN2015039486(ebook)|ISBN 9781119119043(cloth)|ISBN9781119119050(pdf)|ISBN9781119119104 (epub) Subjects:LCSH:Distribution(Probabilitytheory)|Stochasticsequences.| Discretegeometry.|Combinatorialgeometry. Classification:LCCQA273.6.C4532016(print)|LCCQA273.6(ebook)|DDC 519.2/4–dc23 LCrecordavailableathttp://lccn.loc.gov/2015031840 Setin10/12ptTimesLTStd-RomanbySPiGlobal,Chennai,India PrintedintheUnitedStatesofAmerica 10987654321 1 2016 (cid:2) (cid:2) Tothememoryofmyparents AngelosandElpida (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) CONTENTS Preface ix (cid:2) 1 Basicq-Combinatoricsandq-HypergeometricSeries 1 (cid:2) 1.1 Introduction, 1 1.2 q-Factorialsandq-BinomialCoefficients, 2 1.3 q-Vandermonde’sandq-Cauchy’sFormulae, 10 1.4 q-BinomialandNegativeq-BinomialFormulae, 16 1.5 Generalq-BinomialFormulaandq-ExponentialFunctions, 24 1.6 q-StirlingNumbers, 26 1.7 Generalizedq-FactorialCoefficients, 36 1.8 q-Factorialandq-BinomialMoments, 42 1.9 ReferenceNotes, 45 1.10 Exercises, 46 2 SuccessProbabilityVaryingwiththeNumberofTrials 61 2.1 q-BinomialDistributionoftheFirstKind, 61 2.2 Negativeq-BinomialDistributionoftheFirstKind, 66 2.3 HeineDistribution, 69 2.4 HeineStochasticProcess, 73 2.5 q-StirlingDistributionsoftheFirstKind, 77 2.6 ReferenceNotes, 85 2.7 Exercises, 86 (cid:2) (cid:2) viii CONTENTS 3 SuccessProbabilityVaryingwiththeNumberofSuccesses 97 3.1 Negativeq-BinomialDistributionoftheSecondKind, 97 3.2 q-BinomialDistributionoftheSecondKind, 102 3.3 EulerDistribution, 105 3.4 EulerStochasticProcess, 109 3.5 q-LogarithmicDistribution, 114 3.6 q-StirlingDistributionsoftheSecondKind, 117 3.7 ReferenceNotes, 122 3.8 Exercises, 123 4 SuccessProbabilityVaryingwiththeNumberofTrialsandthe NumberofSuccesses 135 4.1 q-PólyaDistribution, 135 4.2 q-HypergeometricDistributions, 143 4.3 Inverseq-PólyaDistribution, 149 4.4 Inverseq-HypergeometricDistributions, 153 4.5 Generalizedq-FactorialCoefficientDistributions, 154 4.6 ReferenceNotes, 164 4.7 Exercises, 164 5 LimitingDistributions 173 (cid:2) (cid:2) 5.1 Introduction, 173 5.2 StochasticandinDistributionConvergence, 174 5.3 LawsofLargeNumbers, 176 5.4 CentralLimitTheorems, 181 5.5 Stieltjes–WigertDistributionasLimitingDistribution, 185 5.6 ReferenceNotes, 193 5.7 Exercises, 193 AppendixHintsandAnswerstoExercises 197 References 235 Index 241 (cid:2)