Univariate Discrete Distributions, Third Edition

Copyright......Page 6
Table of Contents\r......Page 8
Preface......Page 18
1.1.1 Factorial and Combinatorial Conventions......Page 22
1.1.2 Gamma and Beta Functions......Page 26
1.1.3 Finite Difference Calculus......Page 31
1.1.4 Differential Calculus......Page 35
1.1.5 Incomplete Gamma and Beta Functions and Other Gamma-Related Functions\r......Page 37
1.1.6 Gaussian Hypergeometric Functions......Page 41
1.1.7 Confluent Hypergeometric Functions (Kummer’s Functions)......Page 44
1.1.8 Generalized Hypergeometric Functions......Page 47
1.1.9 Bernoulli and Euler Numbers and Polynomials......Page 50
1.1.11 Orthogonal Polynomials......Page 53
1.1.12 Basic Hypergeometric Series......Page 55
1.2.1 Calculus of Probabilities......Page 58
1.2.2 Bayes’s Theorem......Page 62
1.2.3 Random Variables......Page 64
1.2.4 Survival Concepts......Page 66
1.2.5 Expected Values......Page 68
1.2.6 Inequalities......Page 70
1.2.7 Moments and Moment Generating Functions......Page 71
1.2.8 Cumulants and Cumulant Generating Functions......Page 75
1.2.9 Joint Moments and Cumulants......Page 77
1.2.10 Characteristic Functions......Page 78
1.2.11 Probability Generating Functions......Page 79
1.2.12 Order Statistics......Page 82
1.2.13 Truncation and Censoring......Page 83
1.2.14 Mixture Distributions......Page 85
1.2.15 Variance of a Function......Page 86
1.2.16 Estimation......Page 87
1.2.17 General Comments on the Computer Generation of Discrete Random Variables\r......Page 92
1.2.18 Computer Software......Page 94
2.1 LATTICE DISTRIBUTIONS......Page 95
2.2.1 Generalized Power Series Distributions......Page 96
2.2.2 Modified Power Series Distributions......Page 100
2.3.1 Katz and Extended Katz Families......Page 103
2.3.2 Sundt and Jewell Family......Page 106
2.3.3 Ord’s Family......Page 108
2.4.1 Generalized Hypergeometric Probability Distributions......Page 110
2.4.2 Generalized Hypergeometric Factorial Moment Distributions......Page 117
2.5 DISTRIBUTIONS BASED ON LAGRANGIAN EXPANSIONS......Page 120
2.6 GOULD AND ABEL DISTRIBUTIONS......Page 122
2.7 FACTORIAL SERIES DISTRIBUTIONS......Page 124
2.8 DISTRIBUTIONS OF ORDER-k......Page 126
2.9 q-SERIES DISTRIBUTIONS......Page 127
3.1 DEFINITION......Page 129
3.3 MOMENTS......Page 130
3.4 PROPERTIES......Page 133
3.6.1 Approximations......Page 137
3.6.2 Bounds......Page 143
3.6.3 Transformations......Page 144
3.7.1 Computation and Tables......Page 145
3.7.2 Computer Generation......Page 146
3.8.2 Point Estimation......Page 147
3.8.3 Confidence Intervals......Page 151
3.8.4 Model Verification......Page 154
3.9 CHARACTERIZATIONS......Page 155
3.10 APPLICATIONS......Page 156
3.11 TRUNCATED BINOMIAL DISTRIBUTIONS......Page 158
3.12.2 Sums and Differences of Binomial-Type Variables......Page 161
3.12.3 Poissonian Binomial, Lexian, and Coolidge Schemes......Page 165
3.12.4 Weighted Binomial Distributions......Page 170
3.12.6 Correlated Binomial Variables......Page 172
4.2.1 Genesis......Page 177
4.2.2 Poissonian Approximations......Page 181
4.3 MOMENTS......Page 182
4.4 PROPERTIES......Page 184
4.5 APPROXIMATIONS, BOUNDS, AND TRANSFORMATIONS......Page 188
4.6.1 Computation and Tables......Page 191
4.6.2 Computer Generation......Page 192
4.7.1 Model Selection......Page 194
4.7.2 Point Estimation......Page 195
4.7.3 Confidence Intervals......Page 197
4.7.4 Model Verification......Page 199
4.8 CHARACTERIZATIONS......Page 200
4.9 APPLICATIONS......Page 207
4.10.1 Left Truncation......Page 209
4.10.2 Right Truncation and Double Truncation......Page 212
4.10.3 Misrecorded Poisson Distributions......Page 214
4.11 POISSON–STOPPED SUM DISTRIBUTIONS......Page 216
4.12.2 Gamma Distribution......Page 217
4.12.3 Sums and Differences of Poisson Variates......Page 218
4.12.4 Hyper-Poisson Distributions......Page 220
4.12.5 Grouped Poisson Distributions......Page 223
4.12.7 Intervened Poisson Distributions......Page 226
5.1 DEFINITION......Page 229
5.2 GEOMETRIC DISTRIBUTION......Page 231
5.3 HISTORICAL REMARKS AND GENESIS OF NEGATIVE BINOMIAL DISTRIBUTION\r......Page 233
5.4 MOMENTS......Page 236
5.5 PROPERTIES......Page 238
5.6 APPROXIMATIONS AND TRANSFORMATIONS......Page 239
5.7 COMPUTATION AND TABLES......Page 241
5.8.2 P Unknown......Page 243
5.8.3 Both Parameters Unknown......Page 244
5.8.4 Data Sets with a Common Parameter......Page 247
5.8.5 Recent Developments......Page 248
5.9.1 Geometric Distribution......Page 249
5.9.2 Negative Binomial Distribution......Page 252
5.10 APPLICATIONS......Page 253
5.11 TRUNCATED NEGATIVE BINOMIAL DISTRIBUTIONS......Page 254
5.12.1 Limiting Forms......Page 257
5.12.2 Extended Negative Binomial Model......Page 258
5.12.3 Lagrangian Generalized Negative Binomial Distribution......Page 260
5.12.4 Weighted Negative Binomial Distributions......Page 261
5.12.5 Convolutions Involving Negative Binomial Variates......Page 262
5.12.6 Pascal–Poisson Distribution......Page 264
5.12.7 Minimum (Riff–Shuffle) and Maximum Negative Binomial Distributions\r......Page 265
5.12.8 Condensed Negative Binomial Distributions......Page 267
5.12.9 Other Related Distributions......Page 268
6.1 DEFINITION......Page 272
6.2.1 Classical Hypergeometric Distribution......Page 273
6.2.2 Beta–Binomial Distribution, Negative (Inverse) Hypergeometric Distribution: Hypergeometric Waiting-Time Distribution\r......Page 274
6.2.3 Beta–Negative Binomial Distribution: Beta–Pascal Distribution, Generalized Waring Distribution\r......Page 277
6.2.4 Polya Distributions\r......Page 279
6.2.5 Hypergeometric Distributions in General......Page 280
6.3 MOMENTS......Page 283
6.4 PROPERTIES......Page 286
6.5 APPROXIMATIONS AND BOUNDS......Page 289
6.6 TABLES, COMPUTATION, AND COMPUTER GENERATION......Page 292
6.7 ESTIMATION......Page 293
6.7.1 Classical Hypergeometric Distribution......Page 294
6.7.2 Negative (Inverse) Hypergeometric Distribution: Beta–Binomial Distribution\r......Page 295
6.7.3 Beta–Pascal Distribution......Page 297
6.8 CHARACTERIZATIONS......Page 298
6.9.1 Classical Hypergeometric Distribution......Page 300
6.9.2 Negative (Inverse) Hypergeometric Distribution: Beta–Binomial Distribution\r......Page 302
6.10.1 Discrete Rectangular Distribution......Page 304
6.10.2 Distribution of Leads in Coin Tossing......Page 307
6.10.3 Yule Distribution......Page 308
6.10.4 Waring Distribution......Page 310
6.10.5 Narayana Distribution......Page 312
6.11.1 Extended Hypergeometric Distributions......Page 314
6.11.2 Generalized Hypergeometric Probability Distributions......Page 317
6.11.3 Generalized Hypergeometric Factorial Moment Distributions......Page 319
6.11.4 Other Related Distributions......Page 320
7.1.1 Definition......Page 323
7.1.2 Historical Remarks and Genesis......Page 324
7.1.3 Moments......Page 326
7.1.4 Properties......Page 328
7.1.5 Approximations and Bounds......Page 330
7.1.6 Computation, Tables, and Computer Generation......Page 331
7.1.7 Estimation......Page 332
7.1.8 Characterizations......Page 336
7.1.9 Applications......Page 337
7.1.10 Truncated and Modified Logarithmic Distributions......Page 338
7.1.11 Generalizations of the Logarithmic Distribution......Page 340
7.1.12 Other Related Distributions......Page 342
7.2 LAGRANGIAN DISTRIBUTIONS......Page 346
7.2.1 Otter’s Multiplicative Process......Page 347
7.2.2 Borel Distribution......Page 349
7.2.3 Consul Distribution......Page 350
7.2.4 Geeta Distribution......Page 351
7.2.5 General Lagrangian Distributions of the First Kind......Page 352
7.2.6 Lagrangian Poisson Distribution......Page 357
7.2.7 Lagrangian Negative Binomial Distribution......Page 361
7.2.8 Lagrangian Logarithmic Distribution......Page 362
7.2.9 Lagrangian Distributions of the Second Kind......Page 363
8.1.1 Introduction......Page 364
8.1.2 Finite Mixtures......Page 365
8.1.3 Varying Parameters......Page 366
8.2.1 Parameters of Finite Mixtures......Page 368
8.2.2 Parameter Estimation......Page 370
8.2.3 Zero-Modified and Hurdle Distributions......Page 372
8.2.4 Examples of Zero-Modified Distributions......Page 374
8.2.5 Finite Poisson Mixtures......Page 378
8.2.6 Finite Binomial Mixtures......Page 379
8.2.7 Other Finite Mixtures of Discrete Distributions......Page 380
8.3.1 Properties of General Mixed Distributions......Page 381
8.3.2 Properties of Mixed Poisson Distributions......Page 383
8.3.3 Examples of Poisson Mixtures......Page 386
8.3.4 Mixtures of Binomial Distributions......Page 394
8.3.5 Examples of Binomial Mixtures......Page 395
8.3.6 Other Continuous and Countable Mixtures of Discrete Distributions......Page 397
8.4 GAMMA AND BETA MIXING DISTRIBUTIONS......Page 399
9.1 GENERALIZED AND GENERALIZING DISTRIBUTIONS......Page 402
9.2 DAMAGE PROCESSES......Page 407
9.3 POISSON–STOPPED SUM (MULTIPLE POISSON) DISTRIBUTIONS\r......Page 409
9.4 HERMITE DISTRIBUTION......Page 415
9.5 POISSON–BINOMIAL DISTRIBUTION......Page 421
9.6.1 Definition......Page 424
9.6.2 Moment Properties......Page 426
9.6.3 Tables and Approximations......Page 427
9.6.4 Estimation......Page 428
9.6.5 Applications......Page 430
9.7 POLYA–AEPPLI DISTRIBUTION\r......Page 431
9.8 GENERALIZED POLYA–AEPPLI (POISSON–NEGATIVE BINOMIAL) DISTRIBUTION\r......Page 435
9.9 GENERALIZATIONS OF NEYMAN TYPE A DISTRIBUTION......Page 437
9.10 THOMAS DISTRIBUTION......Page 442
9.11 BOREL–TANNER DISTRIBUTION: LAGRANGIAN POISSON DISTRIBUTION\r......Page 444
9.12 OTHER POISSON–STOPPED SUM (MULTIPLE POISSON) DISTRIBUTIONS\r......Page 446
9.13 OTHER FAMILIES OF STOPPED-SUM DISTRIBUTIONS......Page 447
10.1 INTRODUCTION......Page 451
10.2 PROBABILITIES OF COMBINED EVENTS......Page 452
10.3 MATCHING DISTRIBUTIONS......Page 455
10.4.1 Classical Occupancy and Coupon Collecting......Page 460
10.4.2 Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac Statistics......Page 465
10.4.3 Specified Occupancy and Grassia–Binomial Distributions......Page 467
10.5 RECORD VALUE DISTRIBUTIONS......Page 469
10.6.1 Runs of Like Elements......Page 471
10.6.2 Runs Up and Down......Page 474
10.7.1 Early Work on Success Runs Distributions......Page 475
10.7.2 Geometric Distribution of Order k......Page 477
10.7.3 Negative Binomial Distributions of Order k......Page 479
10.7.4 Poisson and Logarithmic Distributions of Order k......Page 480
10.7.5 Binomial Distributions of Order k......Page 482
10.7.6 Further Distributions of Order k......Page 484
10.8 q-SERIES DISTRIBUTIONS......Page 485
10.8.1 Terminating Distributions......Page 486
10.8.2 q-Series Distributions with Infinite Support......Page 491
10.8.3 Bilateral q-Series Distributions......Page 495
10.8.4 q-Series Related Distributions......Page 497
11.1.1 Introduction......Page 499
11.1.2 Tweedie–Poisson Family......Page 501
11.1.3 Negative Binomial Regression Models......Page 503
11.1.4 Poisson Lognormal Model......Page 504
11.1.5 Poisson–Inverse Gaussian (Sichel) Model......Page 505
11.1.6 Poisson Polynomial Distribution......Page 508
11.1.7 Weighted Poisson Distributions......Page 509
11.1.8 Double-Poisson and Double-Binomial Distributions......Page 510
11.1.9 Simplex–Binomial Mixture Model......Page 511
11.2.1 Dandekar’s Modified Binomial and Poisson Models......Page 512
11.2.2 Digamma and Trigamma Distributions......Page 513
11.2.3 Discrete Ades Distribution\r......Page 515
11.2.4 Discrete Bessel Distribution......Page 516
11.2.5 Discrete Mittag–Leffler Distribution......Page 517
11.2.6 Discrete Student’s t Distribution......Page 519
11.2.7 Feller–Arley and Gegenbauer Distributions......Page 520
11.2.8 Gram–Charlier Type B Distributions......Page 522
11.2.9 “Interrupted” Distributions......Page 523
11.2.10 Lost-Games Distributions......Page 524
11.2.11 Luria–Delbrück Distribution\r......Page 526
11.2.12 Naor’s Distribution......Page 528
11.2.13 Partial-Sums Distributions......Page 529
11.2.14 Queueing Theory Distributions......Page 533
11.2.15 Reliability and Survival Distributions......Page 535
11.2.16 Skellam–Haldane Gene Frequency Distribution......Page 540
11.2.17 Steyn’s Two-Parameter Power Series Distributions......Page 542
11.2.18 Univariate Multinomial-Type Distributions......Page 543
11.2.19 Urn Models with Stochastic Replacements......Page 545
11.2.20 Zipf-Related Distributions......Page 547
11.2.21 Haight’s Zeta Distributions......Page 554
Bibliography......Page 556
Index......Page 654