Univariate Discrete Distributions by Norman L.Johnson
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Univariate Discrete Distributions by Norman L.Johnson
This Set Contains:
*Continuous Multivariate Distributions, Volume 1, Models and Applications, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Also
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 Also
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 Also
Discover the latest advances in discrete distributions theory
The Third Edition of the critically acclaimed Univariate Discrete Distributions provides a self-contained, systematic treatment of the theory, derivation, and application of probability distributions for count data. Generalized zeta-function and q-series distributions have been added and are covered in detail. New families of distributions, including Lagrangian-type 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. Also
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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. Also
Beginning with mathematical, probability, and statistical fundamentals, the authors provide clear coverage of the key topics in the field, including:
- Families of discrete distributions Also
- Binomial distribution
- Poisson distribution
- Negative binomial distribution Also
- Hypergeometric distributions
- Logarithmic and Lagrangian distributions
- Mixture distributions
- Stopped-sum distributions Also
- Matching, occupancy, runs, and q-series distributions
- Parametric regression models and miscellanea
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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 computer-based applications. Also
With its thorough coverage and balanced presentation of theory and application, this is an excellent and essential reference for statisticians and mathematicians. Also
Table of Contents
Preface xvii
1 Preliminary Information 1
1.1 Mathematical Preliminaries, 1 Also
1.1.1 Factorial and Combinatorial Conventions, 1
1.1.2 Gamma and Beta Functions, 5
1.1.3 Finite Difference Calculus, 10 Also
1.1.4 Differential Calculus, 14
1.1.5 Incomplete Gamma and Beta Functions and Other Gamma-Related Functions, 16
1.1.6 Gaussian Hypergeometric Functions, 20 Also
1.1.7 Confluent Hypergeometric Functions (Kummer’s Functions), 23
1.1.8 Generalized Hypergeometric Functions, 26 Also
1.1.9 Bernoulli and Euler Numbers and Polynomials, 29
1.1.10 Integral Transforms, 32
1.1.11 Orthogonal Polynomials, 32 Also
1.1.12 Basic Hypergeometric Series, 34
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1.2 Probability and Statistical Preliminaries, 37 Also
1.2.1 Calculus of Probabilities, 37
1.2.2 Bayes’s Theorem, 41 Also
1.2.3 Random Variables, 43
1.2.4 Survival Concepts, 45
1.2.5 Expected Values, 47 Also
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 Also
1.2.10 Characteristic Functions, 57
1.2.11 Probability Generating Functions, 58
1.2.12 Order Statistics, 61 Also
1.2.13 Truncation and Censoring, 62
1.2.14 Mixture Distributions, 64 Also
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 Also
2 Families of Discrete Distributions 74
2.1 Lattice Distributions, 74
2.2 Power Series Distributions, 75 Also
2.2.1 Generalized Power Series Distributions, 75
2.2.2 Modified Power Series Distributions, 79
2.3 Difference-Equation Systems, 82 Also
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 Also
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 Also
2.7 Factorial Series Distributions, 103
2.8 Distributions of Order-k, 105
2.9 q-Series Distributions, 106 Also
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3 Binomial Distribution 108
3.1 Definition, 108
3.2 Historical Remarks and Genesis, 109
3.3 Moments, 109 Also
3.4 Properties, 112
3.5 Order Statistics, 116
3.6 Approximations, Bounds, and Transformations, 116
3.6.1 Approximations, 116 Also
3.6.2 Bounds, 122
3.6.3 Transformations, 123
3.7 Computation, Tables, and Computer Generation, 124 Also
3.7.1 Computation and Tables, 124
3.7.2 Computer Generation, 125
3.8 Estimation, 126 Also
3.8.1 Model Selection, 126
3.8.2 Point Estimation, 126
3.8.3 Confidence Intervals, 130
3.8.4 Model Verification, 133 Also
3.9 Characterizations, 134
3.10 Applications, 135
3.11 Truncated Binomial Distributions, 137
3.12 Other Related Distributions, 140 Also
3.12.1 Limiting Forms, 140
3.12.2 Sums and Differences of Binomial-Type Variables, 140
3.12.3 Poissonian Binomial, Lexian, and Coolidge Schemes, 144
3.12.4 Weighted Binomial Distributions, 149 Also
3.12.5 Chain Binomial Models, 151
3.12.6 Correlated Binomial Variables, 151
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4 Poisson Distribution 156
4.1 Definition, 156
4.2 Historical Remarks and Genesis, 156 Also
4.2.1 Genesis, 156
4.2.2 Poissonian Approximations, 160
4.3 Moments, 161 Also
4.4 Properties, 163
4.5 Approximations, Bounds, and Transformations, 167
4.6 Computation, Tables, and Computer Generation, 170 Also
4.6.1 Computation and Tables, 170
4.6.2 Computer Generation, 171
4.7 Estimation, 173 Also
4.7.1 Model Selection, 173
4.7.2 Point Estimation, 174
4.7.3 Confidence Intervals, 176
4.7.4 Model Verification, 178 Also
4.8 Characterizations, 179
4.9 Applications, 186
4.10 Truncated and Misrecorded Poisson Distributions, 188 Also
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 Also
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 Hyper-Poisson Distributions, 199 Also
4.12.5 Grouped Poisson Distributions, 202
4.12.6 Heine and Euler Distributions, 205
4.12.7 Intervened Poisson Distributions, 205 Also
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5 Negative Binomial Distribution 208
5.1 Definition, 208
5.2 Geometric Distribution, 210 Also
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 Also
5.7 Computation and Tables, 220
5.8 Estimation, 222
5.8.1 Model Selection, 222
5.8.2 P Unknown, 222 Also
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 Also
5.9.1 Geometric Distribution, 228
5.9.2 Negative Binomial Distribution, 231
5.10 Applications, 232 Also
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 Also
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 Also
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 Also
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6 Hypergeometric Distributions 251
6.1 Definition, 251
6.2 Historical Remarks and Genesis, 252 Also
6.2.1 Classical Hypergeometric Distribution, 252
6.2.2 Beta–Binomial Distribution, Negative (Inverse) Hypergeometric Distribution: Hypergeometric Waiting-Time Distribution, 253
6.2.3 Beta–Negative Binomial Distribution: Beta–Pascal Distribution, Generalized Waring Distribution, 256 Also
6.2.4 Pólya Distributions, 258
6.2.5 Hypergeometric Distributions in General, 259 Also
6.3 Moments, 262
6.4 Properties, 265
6.5 Approximations and Bounds, 268
6.6 Tables, Computation, and Computer Generation, 271 Also
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 Also
6.8 Characterizations, 277
6.9 Applications, 279
6.9.1 Classical Hypergeometric Distribution, 279 Also
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 Also
6.10.3 Yule Distribution, 287
6.10.4 Waring Distribution, 289
6.10.5 Narayana Distribution, 291
6.11 Related Distributions, 293 Also
6.11.1 Extended Hypergeometric Distributions, 293
6.11.2 Generalized Hypergeometric Probability Distributions, 296
6.11.3 Generalized Hypergeometric Factorial Moment Distributions, 298 Also
6.11.4 Other Related Distributions, 299
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7 Logarithmic and Lagrangian Distributions 302
7.1 Logarithmic Distribution, 302 Also
7.1.1 Definition, 302
7.1.2 Historical Remarks and Genesis, 303
7.1.3 Moments, 305
7.1.4 Properties, 307 Also
7.1.5 Approximations and Bounds, 309
7.1.6 Computation, Tables, and Computer Generation, 310
7.1.7 Estimation, 311 Also
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 Also
7.1.12 Other Related Distributions, 321
7.2 Lagrangian Distributions, 325
7.2.1 Otter’s Multiplicative Process, 326 Also
7.2.2 Borel Distribution, 328
7.2.3 Consul Distribution, 329 Also
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 Also
7.2.8 Lagrangian Logarithmic Distribution, 341
7.2.9 Lagrangian Distributions of the Second Kind, 342 Also
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8 Mixture Distributions 343
8.1 Basic Ideas, 343
8.1.1 Introduction, 343 Also
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 Also
8.2.1 Parameters of Finite Mixtures, 347
8.2.2 Parameter Estimation, 349
8.2.3 Zero-Modified and Hurdle Distributions, 351
8.2.4 Examples of Zero-Modified Distributions, 353
8.2.5 Finite Poisson Mixtures, 357
8.2.6 Finite Binomial Mixtures, 358 Also
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 Also
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 Also
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9 Stopped-Sum Distributions 381
9.1 Generalized and Generalizing Distributions, 381
9.2 Damage Processes, 386 Also
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 Also
9.6.1 Definition, 403
9.6.2 Moment Properties, 405
9.6.3 Tables and Approximations, 406 Also
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 Also
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 Also
9.13 Other Families of Stopped-Sum Distributions, 426
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10 Matching, Occupancy, Runs, and q-Series Distributions 430
10.1 Introduction, 430 Also
10.2 Probabilities of Combined Events, 431
10.3 Matching Distributions, 434 Also
10.4 Occupancy Distributions, 439
10.4.1 Classical Occupancy and Coupon Collecting, 439 Also
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 Also
10.6.1 Runs of Like Elements, 450
10.6.2 Runs Up and Down, 453 Also
10.7 Distributions of Order k, 454
10.7.1 Early Work on Success Runs Distributions, 454 Also
10.7.2 Geometric Distribution of Order k, 456
10.7.3 Negative Binomial Distributions of Order k, 458 Also
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 Also
10.8 q-Series Distributions, 464
10.8.1 Terminating Distributions, 465
10.8.2 q-Series Distributions with Infinite Support, 470 Also
10.8.3 Bilateral q-Series Distributions, 474
10.8.4 q-Series Related Distributions, 476
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11 Parametric Regression Models and Miscellanea 478
11.1 Parametric Regression Models, 478 Also
11.1.1 Introduction, 478
11.1.2 Tweedie–Poisson Family, 480 Also
11.1.3 Negative Binomial Regression Models, 482 Also
11.1.4 Poisson Lognormal Model, 483
11.1.5 Poisson–Inverse Gaussian (Sichel) Model, 484
11.1.6 Poisson Polynomial Distribution, 487 Also
11.1.7 Weighted Poisson Distributions, 488
11.1.8 Double-Poisson and Double-Binomial 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 Also
11.2.2 Digamma and Trigamma Distributions, 492
11.2.3 Discrete Adès Distribution, 494
11.2.4 Discrete Bessel Distribution, 495 Also
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 Also
11.2.9 “Interrupted” Distributions, 502
11.2.10 Lost-Games Distributions, 503 Also
11.2.11 Luria–Delbrück Distribution, 505
11.2.12 Naor’s Distribution, 507 Also
11.2.13 Partial-Sums Distributions, 508 Also
11.2.14 Queueing Theory Distributions, 512
11.2.15 Reliability and Survival Distributions, 514 Also
11.2.16 Skellam–Haldane Gene Frequency Distribution, 519
11.2.17 Steyn’s Two-Parameter Power Series Distributions, 521 Also
11.2.18 Univariate Multinomial-Type Distributions, 522
11.2.19 Urn Models with Stochastic Replacements, 524 Also
11.2.20 Zipf-Related Distributions, 526
11.2.21 Haight’s Zeta Distributions, 533 Also
Bibliography 535
Abbreviations 631 Also
Index 633 Also
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Author Information
NORMAN L. JOHNSON, PHD, was Professor Emeritus, Department of Statistics, University of North Carolina at Chapel Hill. Dr. Johnson was Editor-in-Chief (with Dr. Kotz) of the Encyclopedia of Statistical Sciences, Second Edition (Wiley). Also
ADRIENNE W. KEMP, PHD, is Honorary Senior Lecturer at the Mathematical Institute, University of St. Andrews in Scotland. Also
SAMUEL KOTZ, PHD, is Professor and Research Scholar, Department of Engineering Management and Systems Engineering, The George Washington University in Washington, DC. Also
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) Also
“The authors continue to do a praise-worthy 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) Also
“These authors have achieved considerable renown for their comprehensive books on statistical distributions.” (Technometrics, August 2006) Also
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Course Features
- Lectures 0
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- Duration 25 hours
- Skill level All levels
- Language English
- Students 99
- Assessments Yes
