Univariate Discrete Distributions, 3rd EditionISBN: 9780471272465
672 pages
August 2005

Description
Continuous Multivariate Distributions, Volume 1, Models and Applications, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson
Continuous Univariate Distributions, Volume 1, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson
Continuous Univariate Distributions, Volume 2, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson
Discrete Multivariate Distributions by Samuel Kotz, N. Balakrishnan and Normal L. Johnson
Univariate Discrete Distributions, 3rd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson
Discover the latest advances in discrete distributions theory
The Third Edition of the critically acclaimed Univariate Discrete Distributions provides a selfcontained, systematic treatment of the theory, derivation, and application of probability distributions for count data. Generalized zetafunction and qseries distributions have been added and are covered in detail. New families of distributions, including Lagrangiantype distributions, are integrated into this thoroughly revised and updated text. Additional applications of univariate discrete distributions are explored to demonstrate the flexibility of this powerful method.
A thorough survey of recent statistical literature draws attention to many new distributions and results for the classical distributions. Approximately 450 new references along with several new sections are introduced to reflect the current literature and knowledge of discrete distributions.
Beginning with mathematical, probability, and statistical fundamentals, the authors provide clear coverage of the key topics in the field, including:
 Families of discrete distributions
 Binomial distribution
 Poisson distribution
 Negative binomial distribution
 Hypergeometric distributions
 Logarithmic and Lagrangian distributions
 Mixture distributions
 Stoppedsum distributions
 Matching, occupancy, runs, and qseries distributions
 Parametric regression models and miscellanea
Emphasis continues to be placed on the increasing relevance of Bayesian inference to discrete distribution, especially with regard to the binomial and Poisson distributions. New derivations of discrete distributions via stochastic processes and random walks are introduced without unnecessarily complex discussions of stochastic processes. Throughout the Third Edition, extensive information has been added to reflect the new role of computerbased applications.
With its thorough coverage and balanced presentation of theory and application, this is an excellent and essential reference for statisticians and mathematicians.
Table of Contents
Preface xvii
1 Preliminary Information 1
1.1 Mathematical Preliminaries, 1
1.1.1 Factorial and Combinatorial Conventions, 1
1.1.2 Gamma and Beta Functions, 5
1.1.3 Finite Difference Calculus, 10
1.1.4 Differential Calculus, 14
1.1.5 Incomplete Gamma and Beta Functions and Other GammaRelated Functions, 16
1.1.6 Gaussian Hypergeometric Functions, 20
1.1.7 Confluent Hypergeometric Functions (Kummer’s Functions), 23
1.1.8 Generalized Hypergeometric Functions, 26
1.1.9 Bernoulli and Euler Numbers and Polynomials, 29
1.1.10 Integral Transforms, 32
1.1.11 Orthogonal Polynomials, 32
1.1.12 Basic Hypergeometric Series, 34
1.2 Probability and Statistical Preliminaries, 37
1.2.1 Calculus of Probabilities, 37
1.2.2 Bayes’s Theorem, 41
1.2.3 Random Variables, 43
1.2.4 Survival Concepts, 45
1.2.5 Expected Values, 47
1.2.6 Inequalities, 49
1.2.7 Moments and Moment Generating Functions, 50
1.2.8 Cumulants and Cumulant Generating Functions, 54
1.2.9 Joint Moments and Cumulants, 56
1.2.10 Characteristic Functions, 57
1.2.11 Probability Generating Functions, 58
1.2.12 Order Statistics, 61
1.2.13 Truncation and Censoring, 62
1.2.14 Mixture Distributions, 64
1.2.15 Variance of a Function, 65
1.2.16 Estimation, 66
1.2.17 General Comments on the Computer Generation of Discrete Random Variables, 71
1.2.18 Computer Software, 73
2 Families of Discrete Distributions 74
2.1 Lattice Distributions, 74
2.2 Power Series Distributions, 75
2.2.1 Generalized Power Series Distributions, 75
2.2.2 Modified Power Series Distributions, 79
2.3 DifferenceEquation Systems, 82
2.3.1 Katz and Extended Katz Families, 82
2.3.2 Sundt and Jewell Family, 85
2.3.3 Ord’s Family, 87
2.4 Kemp Families, 89
2.4.1 Generalized Hypergeometric Probability Distributions, 89
2.4.2 Generalized Hypergeometric Factorial Moment Distributions, 96
2.5 Distributions Based on Lagrangian Expansions, 99
2.6 Gould and Abel Distributions, 101
2.7 Factorial Series Distributions, 103
2.8 Distributions of Orderk, 105
2.9 qSeries Distributions, 106
3 Binomial Distribution 108
3.1 Definition, 108
3.2 Historical Remarks and Genesis, 109
3.3 Moments, 109
3.4 Properties, 112
3.5 Order Statistics, 116
3.6 Approximations, Bounds, and Transformations, 116
3.6.1 Approximations, 116
3.6.2 Bounds, 122
3.6.3 Transformations, 123
3.7 Computation, Tables, and Computer Generation, 124
3.7.1 Computation and Tables, 124
3.7.2 Computer Generation, 125
3.8 Estimation, 126
3.8.1 Model Selection, 126
3.8.2 Point Estimation, 126
3.8.3 Confidence Intervals, 130
3.8.4 Model Verification, 133
3.9 Characterizations, 134
3.10 Applications, 135
3.11 Truncated Binomial Distributions, 137
3.12 Other Related Distributions, 140
3.12.1 Limiting Forms, 140
3.12.2 Sums and Differences of BinomialType Variables, 140
3.12.3 Poissonian Binomial, Lexian, and Coolidge Schemes, 144
3.12.4 Weighted Binomial Distributions, 149
3.12.5 Chain Binomial Models, 151
3.12.6 Correlated Binomial Variables, 151
4 Poisson Distribution 156
4.1 Definition, 156
4.2 Historical Remarks and Genesis, 156
4.2.1 Genesis, 156
4.2.2 Poissonian Approximations, 160
4.3 Moments, 161
4.4 Properties, 163
4.5 Approximations, Bounds, and Transformations, 167
4.6 Computation, Tables, and Computer Generation, 170
4.6.1 Computation and Tables, 170
4.6.2 Computer Generation, 171
4.7 Estimation, 173
4.7.1 Model Selection, 173
4.7.2 Point Estimation, 174
4.7.3 Confidence Intervals, 176
4.7.4 Model Verification, 178
4.8 Characterizations, 179
4.9 Applications, 186
4.10 Truncated and Misrecorded Poisson Distributions, 188
4.10.1 Left Truncation, 188
4.10.2 Right Truncation and Double Truncation, 191
4.10.3 Misrecorded Poisson Distributions, 193
4.11 Poisson–Stopped Sum Distributions, 195
4.12 Other Related Distributions, 196
4.12.1 Normal Distribution, 196
4.12.2 Gamma Distribution, 196
4.12.3 Sums and Differences of Poisson Variates, 197
4.12.4 HyperPoisson Distributions, 199
4.12.5 Grouped Poisson Distributions, 202
4.12.6 Heine and Euler Distributions, 205
4.12.7 Intervened Poisson Distributions, 205
5 Negative Binomial Distribution 208
5.1 Definition, 208
5.2 Geometric Distribution, 210
5.3 Historical Remarks and Genesis of Negative Binomial Distribution, 212
5.4 Moments, 215
5.5 Properties, 217
5.6 Approximations and Transformations, 218
5.7 Computation and Tables, 220
5.8 Estimation, 222
5.8.1 Model Selection, 222
5.8.2 P Unknown, 222
5.8.3 Both Parameters Unknown, 223
5.8.4 Data Sets with a Common Parameter, 226
5.8.5 Recent Developments, 227
5.9 Characterizations, 228
5.9.1 Geometric Distribution, 228
5.9.2 Negative Binomial Distribution, 231
5.10 Applications, 232
5.11 Truncated Negative Binomial Distributions, 233
5.12 Related Distributions, 236
5.12.1 Limiting Forms, 236
5.12.2 Extended Negative Binomial Model, 237
5.12.3 Lagrangian Generalized Negative Binomial Distribution, 239
5.12.4 Weighted Negative Binomial Distributions, 240
5.12.5 Convolutions Involving Negative Binomial Variates, 241
5.12.6 Pascal–Poisson Distribution, 243
5.12.7 Minimum (Riff–Shuffle) and Maximum Negative Binomial Distributions, 244
5.12.8 Condensed Negative Binomial Distributions, 246
5.12.9 Other Related Distributions, 247
6 Hypergeometric Distributions 251
6.1 Definition, 251
6.2 Historical Remarks and Genesis, 252
6.2.1 Classical Hypergeometric Distribution, 252
6.2.2 Beta–Binomial Distribution, Negative (Inverse) Hypergeometric Distribution: Hypergeometric WaitingTime Distribution, 253
6.2.3 Beta–Negative Binomial Distribution: Beta–Pascal Distribution, Generalized Waring Distribution, 256
6.2.4 Pólya Distributions, 258
6.2.5 Hypergeometric Distributions in General, 259
6.3 Moments, 262
6.4 Properties, 265
6.5 Approximations and Bounds, 268
6.6 Tables, Computation, and Computer Generation, 271
6.7 Estimation, 272
6.7.1 Classical Hypergeometric Distribution, 273
6.7.2 Negative (Inverse) Hypergeometric Distribution: Beta–Binomial Distribution, 274
6.7.3 Beta–Pascal Distribution, 276
6.8 Characterizations, 277
6.9 Applications, 279
6.9.1 Classical Hypergeometric Distribution, 279
6.9.2 Negative (Inverse) Hypergeometric Distribution: Beta–Binomial Distribution, 281
6.9.3 Beta–Negative Binomial Distribution: Beta–Pascal Distribution, Generalized Waring Distribution, 283
6.10 Special Cases, 283
6.10.1 Discrete Rectangular Distribution, 283
6.10.2 Distribution of Leads in Coin Tossing, 286
6.10.3 Yule Distribution, 287
6.10.4 Waring Distribution, 289
6.10.5 Narayana Distribution, 291
6.11 Related Distributions, 293
6.11.1 Extended Hypergeometric Distributions, 293
6.11.2 Generalized Hypergeometric Probability Distributions, 296
6.11.3 Generalized Hypergeometric Factorial Moment Distributions, 298
6.11.4 Other Related Distributions, 299
7 Logarithmic and Lagrangian Distributions 302
7.1 Logarithmic Distribution, 302
7.1.1 Definition, 302
7.1.2 Historical Remarks and Genesis, 303
7.1.3 Moments, 305
7.1.4 Properties, 307
7.1.5 Approximations and Bounds, 309
7.1.6 Computation, Tables, and Computer Generation, 310
7.1.7 Estimation, 311
7.1.8 Characterizations, 315
7.1.9 Applications, 316
7.1.10 Truncated and Modified Logarithmic Distributions, 317
7.1.11 Generalizations of the Logarithmic Distribution, 319
7.1.12 Other Related Distributions, 321
7.2 Lagrangian Distributions, 325
7.2.1 Otter’s Multiplicative Process, 326
7.2.2 Borel Distribution, 328
7.2.3 Consul Distribution, 329
7.2.4 Geeta Distribution, 330
7.2.5 General Lagrangian Distributions of the First Kind, 331
7.2.6 Lagrangian Poisson Distribution, 336
7.2.7 Lagrangian Negative Binomial Distribution, 340
7.2.8 Lagrangian Logarithmic Distribution, 341
7.2.9 Lagrangian Distributions of the Second Kind, 342
8 Mixture Distributions 343
8.1 Basic Ideas, 343
8.1.1 Introduction, 343
8.1.2 Finite Mixtures, 344
8.1.3 Varying Parameters, 345
8.1.4 Bayesian Interpretation, 347
8.2 Finite Mixtures of Discrete Distributions, 347
8.2.1 Parameters of Finite Mixtures, 347
8.2.2 Parameter Estimation, 349
8.2.3 ZeroModified and Hurdle Distributions, 351
8.2.4 Examples of ZeroModified Distributions, 353
8.2.5 Finite Poisson Mixtures, 357
8.2.6 Finite Binomial Mixtures, 358
8.2.7 Other Finite Mixtures of Discrete Distributions, 359
8.3 Continuous and Countable Mixtures of Discrete Distributions, 360
8.3.1 Properties of General Mixed Distributions, 360
8.3.2 Properties of Mixed Poisson Distributions, 362
8.3.3 Examples of Poisson Mixtures, 365
8.3.4 Mixtures of Binomial Distributions, 373
8.3.5 Examples of Binomial Mixtures, 374
8.3.6 Other Continuous and Countable Mixtures of Discrete Distributions, 376
8.4 Gamma and Beta Mixing Distributions, 378
9 StoppedSum Distributions 381
9.1 Generalized and Generalizing Distributions, 381
9.2 Damage Processes, 386
9.3 Poisson–Stopped Sum (Multiple Poisson) Distributions, 388
9.4 Hermite Distribution, 394
9.5 Poisson–Binomial Distribution, 400
9.6 Neyman Type A Distribution, 403
9.6.1 Definition, 403
9.6.2 Moment Properties, 405
9.6.3 Tables and Approximations, 406
9.6.4 Estimation, 407
9.6.5 Applications, 409
9.7 Pólya–Aeppli Distribution, 410
9.8 Generalized Pólya–Aeppli (Poisson–Negative Binomial) Distribution, 414
9.9 Generalizations of Neyman Type A Distribution, 416
9.10 Thomas Distribution, 421
9.11 Borel–Tanner Distribution: Lagrangian Poisson Distribution, 423
9.12 Other Poisson–Stopped Sum (multiple Poisson) Distributions, 425
9.13 Other Families of StoppedSum Distributions, 426
10 Matching, Occupancy, Runs, and qSeries Distributions 430
10.1 Introduction, 430
10.2 Probabilities of Combined Events, 431
10.3 Matching Distributions, 434
10.4 Occupancy Distributions, 439
10.4.1 Classical Occupancy and Coupon Collecting, 439
10.4.2 Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac Statistics, 444
10.4.3 Specified Occupancy and Grassia–Binomial Distributions, 446
10.5 Record Value Distributions, 448
10.6 Runs Distributions, 450
10.6.1 Runs of Like Elements, 450
10.6.2 Runs Up and Down, 453
10.7 Distributions of Order k, 454
10.7.1 Early Work on Success Runs Distributions, 454
10.7.2 Geometric Distribution of Order k, 456
10.7.3 Negative Binomial Distributions of Order k, 458
10.7.4 Poisson and Logarithmic Distributions of Order k, 459
10.7.5 Binomial Distributions of Order k, 461
10.7.6 Further Distributions of Order k, 463
10.8 qSeries Distributions, 464
10.8.1 Terminating Distributions, 465
10.8.2 qSeries Distributions with Infinite Support, 470
10.8.3 Bilateral qSeries Distributions, 474
10.8.4 qSeries Related Distributions, 476
11 Parametric Regression Models and Miscellanea 478
11.1 Parametric Regression Models, 478
11.1.1 Introduction, 478
11.1.2 Tweedie–Poisson Family, 480
11.1.3 Negative Binomial Regression Models, 482
11.1.4 Poisson Lognormal Model, 483
11.1.5 Poisson–Inverse Gaussian (Sichel) Model, 484
11.1.6 Poisson Polynomial Distribution, 487
11.1.7 Weighted Poisson Distributions, 488
11.1.8 DoublePoisson and DoubleBinomial Distributions, 489
11.1.9 Simplex–Binomial Mixture Model, 490
11.2 Miscellaneous Discrete Distributions, 491
11.2.1 Dandekar’s Modified Binomial and Poisson Models, 491
11.2.2 Digamma and Trigamma Distributions, 492
11.2.3 Discrete Adès Distribution, 494
11.2.4 Discrete Bessel Distribution, 495
11.2.5 Discrete Mittag–Leffler Distribution, 496
11.2.6 Discrete Student’s t Distribution, 498
11.2.7 Feller–Arley and Gegenbauer Distributions, 499
11.2.8 Gram–Charlier Type B Distributions, 501
11.2.9 “Interrupted” Distributions, 502
11.2.10 LostGames Distributions, 503
11.2.11 Luria–Delbrück Distribution, 505
11.2.12 Naor’s Distribution, 507
11.2.13 PartialSums Distributions, 508
11.2.14 Queueing Theory Distributions, 512
11.2.15 Reliability and Survival Distributions, 514
11.2.16 Skellam–Haldane Gene Frequency Distribution, 519
11.2.17 Steyn’s TwoParameter Power Series Distributions, 521
11.2.18 Univariate MultinomialType Distributions, 522
11.2.19 Urn Models with Stochastic Replacements, 524
11.2.20 ZipfRelated Distributions, 526
11.2.21 Haight’s Zeta Distributions, 533
Bibliography 535
Abbreviations 631
Index 633
Author Information
ADRIENNE W. KEMP, PHD, is Honorary Senior Lecturer at the Mathematical Institute, University of St. Andrews in Scotland.
SAMUEL KOTZ, PHD, is Professor and Research Scholar, Department of Engineering Management and Systems Engineering, The George Washington University in Washington, DC.
Reviews
“With its thorough coverage and balanced presentation of theory and application, this is an excellent and essential reference for statisticians and mathematicians.” (Xolosepo, 27 October 2012)
"The authors continue to do a praiseworthy job of making the material accessible in the third edition. This book should be on every library's shelf." (Journal of the American Statistical Association, September 2006)"These authors have achieved considerable renown for their comprehensive books on statistical distributions." (Technometrics, August 2006)
"Encyclopedic in nature, the book continues to be a valuable reference." (Mathematical Reviews, 2006d)
"This is an important book that should be part of every statistician's library." (MAA Reviews, January 2, 2006)
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