636 INDEX. exponential, 210 normal, 89, 475 Pareto, 527 rectangular (uniform), 97, , 515 Student s t, 87, 498 Weibull, 510

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1 Index Added zeroes, 194, 318, Analysis of contingency tables, 186 Applications acceptance sampling, 136 accidents, , 232, 283, , 419 actuarial, 360, 366, 432, 486 animal abundance, 136, 212, 281, 386 archaeology, 489 attendances, 526 audit sampling of accounts, 300 baseball, 245 bank failures, 478 bibliographic (library), 232, 280 biometry, 188, 232, 282, 356, 400 birds, 316, 478 boating trips, 488 bosons, 445 bus drivers, 419 capture recapture (mark-release) sampling, 136, 188, , 526 cancer studies, 188, 233 caries prevention, 354 carrier-borne epidemics, chemical kinetics, 297 chromosomes, 283 city sizes, 533 colonies of bacteria, 188 Common Birds Census, 478 computer storage analysis, 430, 444 congenital malformations, 212 consumption of alcohol, 447 coronary heart disease, 478 correct and incorrect keys, 285 cosmic ray counts, 440, 443 dams, 140 dentistry, 280 deaths of elderly women, 357 deaths from horse kicks, 157 demography, 138, 187, 212, , 281, 297 diamond deposits, 486 dice data, 147 disease incidence, 188, , 283 disease states, 280, 347, 505 dissortative mating, 455 DNA, 136, 188, 233, 430 dose response, 204, 443 drug abuse, 243 drug receptor, 140 ecology, 209, , 414, 458 economics, 188, 232, 347 entomology, 151, 232, 311, 316, 430 epidemiology, 151, 280, 443, 446 exceedances, , 281 family size/composition, 152, 235, 282, 490 fetal deaths, 283 fermions, 445 fisheries research, 347 filing systems, 526 flaws in cloth, 232, 240, 283 gambling, 430 genetics, , 246, 283, , 430, 438, 443, , , 526 hemocytometer data, 212, 241 high explosive shells, 526 image analysis, 343, 348, 350 industrial absenteeism, 419 inspection sampling, 232, 281, 300 Univariate Discrete Distributions, Third Edition. By Norman L. Johnson, Adrienne W. Kemp, and Samuel Kotz Copyright 2005 John Wiley & Sons, Inc. 633

2 634 INDEX Applications (Continued) insurance claims, 233, 242, 250, 369, 486, 520, 527 inventory control, 261, 316, 509 inventory decision theory, 99 jury decisions, 359 larvae, 195, 243, 368, , , 414, 416, 418 leads in coin tossing, learning processes, 526 leucocyte data, 397, 400 light quanta, 198 linguistics, 511, 528, 530 lost sales, 232 machine maintenance, 140, 508 market research, 141, 232, 282 medical trials, 282, 526 meteorology, 212, 458, 518 molecular biology, 458 musicology, 511 mutation of bacteria, 441, oil exploration, 470, 472 oviposition, oxidation of rubber, 441 parasite data, 316, 467 photon counting, 243 plant ecology, 163, , 316 plastic intrusion molding, 467 point quadrat data, 282 polymer reactions, 443 population counts, 339 predator prey, 443 proofreading, 465 proportion of sterile couples, 474 psychology, 232, 430, 458 purchasing and repeat buying, 232, , 311, 316, 318, , 486 pulmonary fibrosis, 155 quadrat data, , 316, 400, 409 quality control, 186, 279, 458 quantum physics, 186, 468, 470 queues, 187, 212, 423, , radioactivity, 158, 502 random mappings, 336 rare animals, 465 residence of animals, 354 retail food stores, 410 runs of plant species, 211 safety campaign, 525 scientific productivity, 528 sedimentology, 347 seed germination, 282, 359 sentence length, 486 sizes of firms, 531 species abundance, 303, 486 species per genus, 287, 531 speech recognition, 348 spread of ideas, 339 start-up testing, 458 statistical physics, 430, , 470 sterile couples, 232 storm duration, 324 stressful events, 478 sudden infant deaths, 478 suicides, 157 surnames, 528 takeover bids, 478, 488 telephony, 140, 187, 242 thermodynamic processes, 339, 528 time to extinction, 472 toxicology, 153, 282 toxoplasmosis, 490 traffic studies, 188, 282 twin studies, 281 two-by-two contingency tables, 280 weapon defense, 147, 467 weather cycles, 212 word associations, 534 word-frequency data, 297, 486 Approximations Camp Paulson, 119, 219 Gram Charlier, , 219 Laplace, Peizer and Pratt, 118, 168, , 271 Arrangements around a circle, 452 Bailey Slater definition, 35 Banach s matchbox problem, 245 Bathtub failure rate, 517 Bayesian inference, 41 42, 66, 223, 245, 273 diffuse (uninformative, vague) prior, 42 informative prior, 42 posterior odds, 42 prior/posterior probability, 42 Bayesian interpretation of a mixture, 347 Bayes postulate, 42 Bayes theorem, 41 Bernoulli variables, 108 Beta-related distributions beta-binomial, 87 88, beta-correlated binomial, 150 beta mixture of Poissons, 368 beta-negative binomial I (beta Pascal), 87 88, beta-negative binomial II, 379

3 INDEX 635 Binomial distribution, 75 76, 81 82, 84, 86 88, 90, , 99, 105, , 515 applications, approximations, bounds, and tables, arc sine transformation, 124 Bayesian inference, Bernoulli distribution, 108 beta (F ) relationship, 78, 113 binomial mixing of binomials, 374 binomial mixing of Poissons, 371 binomial waiting-time distribution, 209 chain binomials, characterizations, computation, tables, and computer generation, confidence intervals, correlated binomial rv s, estimation, genesis and history, 109 as a limiting distribution, 136 limiting forms, 140 mixed binomial distributions, , , model selection, 84, 88, 126 model verification, moment-type properties, negative binomial tail relationship, order statistics, 116 other properties, positive binomial, 111, 137 pseudo-binomial variables, standardized variable, 114 sums and differences of binomial rv s, transformations, truncated, use in McNemar s test, 136 use in sign test, 136 weighted binomial, zero-modified, 354 Binomial expanding and splitting, 182 Binomial theorem, 3 Birthday problem, 440 Boole Bonferroni Fréchet inequalities, Boole s formula, 39 Bose Einstein statistics, Calculus of probabilities, Censoring methods, 63 Chain binomial models Greenwood, Reed Frost, Characteristic function (cf), 32, 50, 57 58, 61 Characterizations, 45, 52, 57, 62, 76, 81 binomial distribution, damage process, geometric distribution, , hypergeometric distributions, logarithmic distribution, negative binomial distribution, Poisson distribution, Clumping (clustering), 195, 382, 393, , , Coincidences, Committee problems, 442 Complement of an event, 38 Complete family, 68 Composition, 196 Compounding, 344, 361, 382 Computer generation, acceptance complement method, 310 acceptance rejection method, 126, 272 alias method, 72, 222, 310 beta (median) method, 125 binomial rv s, build-up/chop-down search, 172 coin-flip (yes-no) method, 126 composition methods, 172 discrete thinning, 310 distribution nonspecific method, distribution specific method, 72 dynamic thinning, 310 envelope distribution, 272 exponential gap method, 172 frequency table method, 72, 222 hypergeometric rv s, 272 indexed table look-up method, 72, 222 inversion method, logarithmic rv s, 304, 310 negative binomial rv s, period of a generator, 71 Poisson rv s, pseudo-random number algorithm, 71 search-from-origin method, 172 table look-up method, 72, 222 target distribution, 72 uniform random-number generators, 125 Walker s alias method, 72 Computer software, 73 Conditional probability, 40 41, 47 Confidence intervals, 71 Consecutive k-out-of-n :F failures, 462 Contagion model, 382 Continuity correction,

4 636 INDEX Continuous distributions beta (F ), 113, 212, , 218, 235, 245, , , chi-square, , 220, 243, 270, exponential, , , 221, 228, gamma, , inverse Gaussian, 366, 373 lognormal, 370, noncentral chi-square, 243 normal, , 132, 140, 196, , 240 Pearson Type III, 370 Continuous mixtures of discrete distributions, of binomials, of negative binomials, of Poissons, , 304 Continuous random variable, 43 Convolution of random variables, 60, 361 binomials, , Poisson multiplets, 196 pseudobinomials, 143 Correlation coefficient, 56 Countable mixtures of discrete distributions, Coupon collecting (dixie-cup/golliwog) problem, 97, Covariance, 56 Cumulants, 54 56, 77, 81, 94, 98 Cumulative distribution function (cdf), 43 Damage process, , Data sets with a common parameter, Decomposable, 58 Decomposition of a distribution, 345 De Finetti s theorem, 389 Degrees of belief, 41 Delta method, 65 De Moivre Laplace theorem, 114 De Morgan s laws, 39 Derangements, 4 Differences of zero, 12 Differential calculus, Differential operators, D and θ, 14 Discrete analogues of continuous distributions Adès, exponential, 210 normal, 89, 475 Pareto, 527 rectangular (uniform), 97, , 515 Student s t, 87, 498 Weibull, 510 Discrete distribution, 43 Discrete distributions, see also Beta-related distributions; Binomial; Convolution of rv s; Discrete analogues of continuous distributions; Distributional types; Distributions of order k; Families; Finite mixtures; Generalized hypergeometric; Geometric; Hermite; Hypergeometric; Lagrangian, Lagrangian Poisson; Logarithmic; Log-series; Mixtures; Negative binomial; Negative hypergeometric; Neyman Type A; Occupancy; Poisson; Poisson-binomial; Poisson Pascal; Poisson-related; Pólya Aeppli; Power series; q-series; Queueing theory; Stopped-sum; Stuttering; Zero modified; Zero truncated Abakuks, 297 absorption, Adès, Anderson, 97 Arfwedson, asynchronous counting distribution, 246 Bailey Daum, 474 Beall and Rescia, 93, Bell, 103 Bernoulli, 108, 136, , Beta-binomial, 87 88, , 374 Beta-correlated binomial, 480 Beta-geometric, 237, 239 Beta-negative-binomial, Beta-Poisson, BF3-P, 319 binomial-beta, 480 binomial distribution of Poisson, 144 binomial waiting-time distribution, 209 Borel, 100, 327, 328 Borel Hermite, 400 Borel Tanner, 196, 288, , Bose Einstein, burnt-fingers, 203 Chung Feller, 97, classical matching, 97, condensed negative binomial, confluent hypergeometric, 202, 214 Consul, 100 Consul s generalized Poisson, 196, Coolidge, correlated binomial, , 480 Dacey, 297 Dandekar s modified binomials, Dandekar s modified Poissons, 492

5 INDEX 637 Darwin, 297 Delaporte, 242 digamma/trigamma, 297, discrete Adès, discrete Bessel, discrete lognormal, , discrete Mittag Leffler, discrete normal, 89, 475 discrete Pareto, 527, 532 discrete Pearson III, 532 discrete rectangular (uniform), 88, 97, , 460 discrete Student s t, 87, 498 discrete Weibull, , displaced Poisson, 93, double binomial, 480, 490 double Poisson, 480, 489 Ehrenfest heat exchange, 297 Engen s extended negative binomial, , , 482 Euler, 205, 325, , 475, 477 Ewens, EXBERT, 154 exceedances, 94, , 281 extended hypergeometric, 85, extended Stirling series, Estoup, 528 Feller Arley, 144, 499 Feller Shreve, 237 Fermi Dirac, 445 Fibonacci, 455 Fisher s modified Poisson, 398 Fréchet s matching, Furry, 210 Galton Watson, 499 Geeta, 100, Gegenbauer, 144, 399, generalized Eulerian, 103 generalized exponential, 214 generalized Gegenbauer, generalized Markov Pólya, 299 generalized Pólya Aeppli, 414 generalized poly nacci, 455 generalized Waring, geometric Poisson, 412 GG1 distribution, 319 Gold and Gerstenkorn s generalized Poisson, 203 Good s distributions, 297, , 532 Gram Charlier Type B, Grassia-binomial, I, II, grouped Poisson, Gumbel, 97, 436 Gurland s generalized Neyman Type A, Haight s accident, 99 Haight s harmonic, 534 Haight s insurance claims, Haight s Lagrangian, 326 Haight s zeta, Harkness, Heine, 205, 466, 468, , 475 Hurwitz, 530 hyperbinomial, 296 hyperlogarithmic, 295, 320 hypernegative binomial, 85, hyper-poisson, 85, 87, 93, imperfect inspection hypergeometric, 300 intervened Poisson, intrinsic hypergeometric, 301 inverse binomial, 505 inverse hypergeometric, 93, , inverse Markov Pólya, 93, 257 Ising Stevens, 94, Jackson, 469 Katti Type H 1, 368 Katti Type H 2, 97, 214, 298, 379 Kemp s limited risk c P p,99 Kempton s log-series generalization, 319 Kirkman, 292 Kulasekera Tonkyn, , 531 Laplace Haag matching distribution, 97, 435 leads in coin tossing, Lexian, 148 linear-fractional, 144 logarithmic-poisson, 346 log-zero, 318 log-zero Poisson (LZP), 372, lost-games, 296, , Lotka, 297, 528 Luria Delbrück, Lüders Formel I, II, 241 Markov Pólya, 299 Marlow s factorial, 105, maximum negative binomial, Maxwell Boltzmann, 444 minimum negative binomial, Mittag Leffler, 237, Möbius, 144 Moran Gani, 295 Morlat s generalized Poisson, 202 Morse, 449 Morse balking distribution, 473 Naor, Narayana,

6 638 INDEX Discrete distributions (Continued) negative hypergeometric, 87, 93, 97, , , 519 Neyman Types B and C, 93, noncentral hypergeometric, 299 noncentral negative binomial, number of inversions, 476 Ong, 319 Palm Poisson, 297 Pascal, see Negative binomial distribution Pascal Poisson, , Poisson-beta, 368 Poissonian trials, 145 Poisson*binomial, 242 Poisson-binomial, 371, 374 Poisson s exponential binomial limit distribution, 509 Poisson Hougaard (cluster-size), 237 Poisson-inverse Gaussian, 366, 373, 479, Poisson Lexis, 149 Poisson Lindley, 370 Poisson logarithmic, 346, 372 Poisson lognormal, 370, 479, Poisson mixture of binomials, 374 Poisson mixture of negative binomials, 367 Poisson mixture of Poissons, see Neyman Type A Poisson*negative binomial, 242 Poisson Pascal, Poisson rectangular, 368 Poisson-stopped sums, Poisson truncated gamma, 369 Poisson truncated normal, 370, 398, 483 Pollaczek Geiringer, 393 Pólya, 93, 97, 213, Pólya Eggenberger, 93, 213 polylogarithmic, 325 poly nacci, 455 pseudo-euler, 471 quasi-binomial, I and II, 103 quasi-hypergeometric, I and II, 102 quasi-pólya, I and II, 102 Quinkert, q-deformed binomial, record value distributions, Riemann zeta, 527 Roy s discrete normal, 498 riff shuffle, Rogers Szegö, 469 runs up and down, Salvia-Bollinger, 237, 511 Sibuya, 237 short, , 520 Sichel distribution, Skellam Haldane, 424, standard Zipf, 532 Stevens Craig, 97, Steyn s two-parameter power series distribution, Stieltjes Wigert, 469 Stirling of the first kind, 93, 315, Stirling of the second kind, 93, , 206 strict arcsine, 250 stuttering Poisson, 195, 393 sub-poisson, 200, 214 Subrahmaniams generalized Neyman, strict arcsine, 250 super-poisson, 200 Tanner Borel, 100, 196, 288, , Thomas, 93, univariate multinomial-type, Wall, 469 Waring, 94, , 517, 519 weighted binomial, weighted negative binomial, 240 Weiss Dietz Downton, Yousry Srivastava, 297 Yule, 94, , 331, 517, zeta, 515 Zipf, Zipf Mandelbrot, 530 Discrete random variable, 43 Dispersion, over-, under-, equi-, 70, 94, 98, 203, 216, 361, 365 Displacement operators (backward, central, forward), Distributional types ascertained, Bissinger (STER), 99, 257, 261, clustered (compounded), 382 conditional, 44, 115, 135, 166, 174, 176, 178, , deformed, discrete linear exponential, 75, 108, 210 extended, factorial series, 82, geometric-tail distributions, 450 group size, 233, 305, 505 hurdle, Hurwitz, 530 inflated/deflated, 194, 351

7 INDEX 639 interrupted, intervened, Katz, 82 85, 519 lattice, maximum entropy distributions (MEDs), 475, 531 misrecorded, mixing, , mixture, 64, modified power series, order k, see Distributions of order k parent distribution, 508 par grappes, 195 partial sums, 508 positive, 174 power series, proper distribution, 43 queueing theory, reliability/survival, renewal, 512 runs, size-biased, success runs, 455 truncated, 63, , , , waiting time, 209, 281 weighted, 94 95, , 240, 294, 308 Distribution function, see Cumulative distribution function (cdf) Distributions of order k, , 301, binomial of order k, I, II, III, compound Poisson of order k, 455, 460 extended geometric of order k, 457 Fibonacci, 455 further distributions of order k, geometric of order k, intervened geometric of order k, 457 logarithmic of order k, 455, 461 Markov-geometric of order k, 457 negative binomial of order k, I, II, III, 455, Poisson of order k, 455, poly nacci, 455 success-runs distributions of order k, 455 Diversity, 303, 310, 316 Ehrenfest model of heat exchange, 525 Elementary functions generalized hypergeometric representations, 28 hypergeometric representations, 22 EM algorithm, 350 Estimable parameters of finite mixtures, Estimation Bayesian methods, 66 even-points method, 398 first-moment equation, 70, 78 generalized/modified/partial/penalized, maximum-likelihood methods, 70 maximum likelihood, 68 69, mean and zero-frequency method, 224 method of moments, 69 70, 224 minimum chi-square method, 85, 227 minimum variance unbiased, 68, 78, 80, 223 pseudo-likelihood methods, 70 Estimator, see also Estimation asymptotically efficient, 68 asymptotically unbiased, 67 biased/unbiased, 67 consistent, 67 efficient, 68 minimum variance unbiased, 68 relative efficiency, 67 sufficient, 68 Eulerian integral of the first kind, 8 Euler s constant, 6, 9 Euler s hypergeometric integral, 21 Euler transformations, 22 Everett s central difference formula, 11 Exchangeable events, 431 Expansions Barnes, 8 binomial, 3 Camp Paulson, 119, 219 Edgeworth, 167 Faà di Bruno, 383, 461 Gram Charlier, , 219 Lagrange, 15, 80 multinomial, 4 Stirling, 7 Waring, 257, 289 Expected value, Factorial cumulants, 55, 77 Factorial moments, 52 54, 76 77, , 440, Failure rate (FR), 45 46, 114, 165, 218, Families of distributions, Abel, Bissinger (STER) distributions, 261 Crow and Bardwell, 85 difference equation, discrete linear-exponential, 75, 108 discrete Pearson, 82 Efron s double exponential, 480 exponential family, 156

8 640 INDEX Families of distributions (Continued) extended Crow and Bardwell, 85 extended Katz, factorial series distributions, generalized factorial series distributions, 105 generalized power series distributions (GPSD), Gould, Hurwitz zeta, Katz, Kemp s hypergeometric factorial distributions, 96 99, Kemp s hypergeometric probability distributions, 89 96, Kemp s hypergeometric recast distributions, 99 Khatri and Patel Types A, B and C, Lagrangian expansion, Lerch, maximum entropy, 475, 531 modified power series distributions (MPSD), multiparameter power series distributions, 80, Ord, order-k, Patil s two-parameter power series distributions, 79 Poisson Katz family, Poisson polynomial, 480, Poisson power series distributions, 393 power series distributions (PSD), q-series, , Riemann zeta, simplex-binomial, 480, Steyn s two-parameter PSD, 80 stopped-sum distributions, , 461 Sundt and Jewell, Tweedie Poisson, weighted Poisson regression family, 480, Fermi Dirac statistics, Finite difference calculus, Finite mixtures of discrete distributions, 344, applications, binomials, other distributions, Poissons, Fisher s exact test, 280 Force of mortality, 515 Fourier transform, 32 Fréchet inequalities, Functions (mathematical) basic hypergeometric, Bessel, 19, 25, 28 beta, 5 9 bilateral q-series, 37 confluent hypergeometric, digamma, trigamma, 8 error functions, 18, 25 gamma, 5 9 Gaussian hypergeometric, generalized hypergeometric, generalized Riemann zeta, 19 Hermite, modified Hermite, 25 Horn Appell functions, 28 incomplete gamma, beta, Lerch, 20 Kummer, 23 normal distribution, 18 psi, 8 q-series, 34 37, 464 Riemann zeta, Gasper Rahman definition, Gaussian (basic) binomial coefficient, 35 Gauss s multiplication theorem, 7 Gauss s summation theorem, 21 Generalized distributions, 361, 382 Generalizing distributions, 361, 382 Generalized hypergeometric distributions, , Generalized variance, 67 Generating functions, central (corrected) moment, 50 51, 61, 76 cumulant, 54, 76 exponential, 443 factorial cumulant, 55, 76 factorial moment, 54, 76, 83 84, 92, 96 moment (uncorrected), 50 probability, Geometric distribution, 93, , 304, 308, , 515, 519, 529, 531 applications, characterizations, , computer generation, 221 estimation, 211 exponential relationship, 210 genesis, infinite divisibility, 211 Markovian (no-memory) property, 210, 449 moment and other properties, 210 order statistics, 211 truncated, zero-modified, 354

9 INDEX 641 Goodness-of-fit, 66, , 179, 227 Greenwood chain binomial model, Gurland s theorem, 364, 383 Half-mode, 51 Hazard function, 45 accumulated, 45 cumulative, 45 Hazard rate (increasing/decreasing), 46 Heine s theorem, 36 37, 467, 468 Hermite distribution, 93, 374, , 483 estimation, extended, 399 genesis, 394, moment and other properties, 396 probabilities, Heterogeneity, 382 Hidden Markov chain, 350 Homoscedasticity, 57 Hypergeometric distribution, 87 88, 93, 97, 108, , 121, , 140 applications, approximations and bounds, beta-binomial, , 374 beta-binomial regression model, 282 beta-negative binomial (beta-pascal), , 376 characterizations, classical, comparison with other distributions, 264 computation, tables, and computer generation, estimation, extended hypergeometric, extended negative hypergeometric, genesis and history, generalized hypergeometric factorial distributions, generalized hypergeometric probability distributions, generalized Markov Pólya, 299 generalized Waring, hypergeometric mixture of binomials, 375 imperfect inspection, intrinsic hypergeometric, 301 inverse (negative) hypergeometric, 93, 97, , limiting, mixed hypergeometric distributions, moment properties, noncentral hypergeometric, 299 other properties, Pólya, positive hypergeometric, tail properties, Types I, II, III and IV, urn models, 252, , 281 waiting-time distribution, , 281 weighted hypergeometric, 262 Hypothesis testing, 66 Identifiability of mixtures, , 361 Inclusion exclusion principle, 39 Incomplete beta function ratio, 17 18, 212, 218, 235, 245 Incomplete gamma function ratio, 16 17, 165, 171, 220 Independent events, 40 Index of clumping, 403 of dispersion, 163, , 306 of diversity, 303, 310, 316 of kurtosis, 51 of skewness, 51 Indicator variable, 432 Indistinguishable particles, Inequalities Bienaymé Chebyshev, 49 Cauchy Schwartz, 49 Chebyshev, 49 Cramér Rao, 68 Jensen, 49 Markov, 49 Infinite divisibility, 58, 164, 195, 211, 218, 308, geometric infinite divisibility, 389 Infinite mixtures of discrete distributions, Integral transforms exponential Fourier, 32 Laplace, 18, 32 Mellin, 32 Interval estimates, Inversion formulas for cf s, Joint cumulative distribution function, 44 Joint distribution, 44 Joint moments and cumulants, 56 Joint probability density function, 44 Joint probability generating function, 60 k-combinations C(n,k), P(n,k), C R (n, k), P R (n, k), 4 k-component finite mixtures, , , Kolmogorov s approximation, 121 Kronecker delta, 13

10 642 INDEX Kummer theorems, 24 Kurtosis, 51 Lagrangian distributions, 82, , basic of the first kind (BLD 1 ), 100 basic of the second kind (BLD 2 ), 101 Borel, 100, 327, Consul, 100, 327, delta of the first kind (DLD 1 ), 100 delta of the second kind (DLD 2 ), 101 Geeta, 100, 327, general of the first kind (GLD 1 ), 100, general of the second kind (GLD 2 ), 101, 342 Haight, 326 Katz, 100, 331 Lagrangian logarithmic, 101, Lagrangian negative binomial, 101, Lagrangian Poisson, 101, 196, , 424 lost-games, Otter s multiplicative process, Tanner Borel, 196, 288, , Landau s notation, 15 Laplace transform, 18, 362 Lattice distribution, 74 Law of small numbers, 157 Legendre s duplication formula, 7 Lévy s existence condition, 394 Lévy s theorem, 363 Lexis ratio, 149 L Hôpital s rule, 15 Likelihood, 42, Logarithmic distribution, 75 76, 88, 93, 213, 215, 218, 221, , 236, 238, 288, 295, , 515, 529 applications, approximations and bounds, 309 characterizations, computation, tables, and computer generation, 310 entropy, 308 estimation, Fisher s derivation, 238, genesis and history, generalizations, infinite divisibility, 308 logarithmic mixture of binomials, 375 logarithmic mixture of Poissons, 372 logarithmic Poisson, 346 model selection, 311 modified, 318 moment properties, other properties, related distributions, truncated, weighted, 308 with zeros, 318, , 355 Log-concavity/log convexity, 43, Logical product (intersection), 38 Logical sum (union), 38 Log-linear odds, 147, 467 Log-linear probabilities, 147, 467 Log-series distribution, , 315 Maceda s theorem, 364 Marginal distribution, 44 Markovian property, , 218, 228 Markov random field, 350 Matches (coincidences), 90, classical matching distribution, 97, Fréchet matching distribution, 97, Gumbel matching distribution, 97, 436 k-pack matching problems, Laplace Haag matching distribution, 97, random splitting, 439 two-pack matching problems, Maximum likelihood equations, 69 Maxwell Boltzmann statistics, 186, Mean residual life function, Mean squared error, 67 Median, 51, 61 Median class, 61 Mellin transform, 32, 378 Method of statistical differentials, 65 Mill s ratio, 114 Mixing distribution, 64, beta, 99, , 253, 256, 262, 272, 288, 290, 298, beta-type, 94, 380 binomial, 277 gamma, 99, , 215, 222, 231, 240, 247, 262, 346, gamma-type, 94, Poisson, 366, 367, 374 Mixtures of discrete distributions, 64, applications, , Bayesian interpretation, 347 binomials, 298 continuous and countable mixtures, , finite binomial mixtures, finite mixtures, , , finite Poisson mixtures, identifiability, 349, 357, 361 infinite binomial mixtures, infinite Poisson mixtures, k-component mixture, 348

11 INDEX 643 mixtures with varying parameters, 345 negative binomials, 298 notation, 345, other infinite mixtures, parameter estimation, zero-modified and hurdle distributions, Modal cumulative probability, 169 Modal probability, 118, 169 Mode, 51 Moments, absolute, 52, 111 ascending (rising) factorial, 53 central (corrected, about the mean), central product, 56 coefficient of variation, 51 descending (falling) factorial, 52 54, exponential moments, 521 inverse (negative) factorial, 111 mean, 50 mean deviation, 52 moment ratios, product moments, 56 seminvariants, 54 standard deviation, 51 uncorrected (about zero), 50 variance, 51 Multimodality, 51 Multinomial theorem, 4 Mutually exclusive, 38 Mutually exhaustive, 40 Negative binomial, 75 76, 81 82, 84, 86 88, 90, 93, 95, 97, 102, , 346, 366, 479 applications, approximations, arc sinh transformation, 220 as a limiting distribution, 209, 213, 236 as a regression model, beta relationship, 78, binomial tail relationship, binomial waiting-time distribution, 209, 221 chain negative binomial model, characterizations, computation, tables, and computer generation, condensed negative binomial, convolutions, displaced, 235 estimation, extended, gamma mixed Poisson model, genesis and history, hurdle model, 248 hyper-negative binomial, infinite divisibility, 211 limiting forms, 212, logarithmic relationship, 213 maximum negative binomial, minimum negative binomial (riff-shuffle), mixtures of negative binomials, 214, , model selection, 222 moment properties, noncentral, order statistics, 211 other properties, parameterizations, , Pascal distribution, 209, 515 recent estimation developments, stochastic processes, 213, 215 stuttering negative binomial, 400 truncated, weighted, 240 zero-modified, 248 Negative hypergeometric distribution, 87, 93, 97, , , generalized form, 297, 298 Newton s forward difference formula, 10 Neyman Type A distribution, 93, 366, applications, approximations and tables, estimation, generalizations, genesis, 403 limiting forms, 406 moments, probabilities, Types B and C, 93, 417 Normalizing transformation, 124, 168, 220 Numbers Bernoulli, binomial, 2 Euler, Gaussian binomial, 35 Stirling of the first and second kinds, Numerical differentiation formula, 14 Occupancy distributions, Order statistics, 61 62, 116, 166, 211, 231 Ord s method of plotting, 88, 126, 173, 222 Parameter, 45, 67 Parameter summed or integrated out, Parametric regression models, negative binomial regression, Poisson regression, 479

12 644 INDEX Parzen s axioms, 159, 393 Pauli exclusion principle, 444 Poisson-binomial distribution, 93, estimation, genesis, 400 modality, 401 moment and other properties, probabilities, 401 Poisson distribution, 75 76, 80 82, 84, 86 88, 90, 93 94, 97, 99, , , , 215, 218, 222, 262, 264, 434, 479, 515 applications, approximations and bounds, as a limiting distribution, axiomatic approach, Bayesian estimation, characterizations, chi-square (gamma) relationship, 78, 158, 165, 167, computation, tables and computer generation, confidence intervals, difference of two rv s, displaced, double truncation, estimation, genesis and history, grouped, infinite divisibility, 164 left and right truncation, limiting forms, 166, misrecorded distribution, mixed Poisson distributions, , , modal probability approximations, 169, 170 model selection, model verification, moment-type properties, order statistics, 166 other properties, Poisson process, 95, Poisson regression, 188, 227 positive Poisson distribution, 174 Rao Rubin theorem, renewal counting process, 158 square-root transformation, 168 standardized variable, 166 sums of rv s, zero deflated, 194, zero inflated (with zeros), 194, zero truncated, 174, 176, 194, 480 Poisson Pascal distribution, 387, 411, Poisson-related distributions clustered, composed, compound, 195 Consul s generalized (Lagrangian) Poisson, 196 Euler, 205 Fisher s modified Poisson, 398 gamma mixture of Poissons, generalized Poissons, 195 Gold and Gerstenkorn s generalized Poisson, 203 grouped, Heine, 205 hyper-poisson, 85, 93, intervened Poisson, modified, Morlat s generalized Poisson, 202 par grappes, 195 Pascal Poisson, , Poisson-binomial, 374, Poisson s exponential binomial limit, 509 Poisson inverse Gaussian, 366, 373, 479, Poisson Katz, Poisson lognormal, 370, Poisson multiplet, 196 Poisson-negative binomial, Poisson*negative binomial (Delaporte), 242 Poisson Pascal, Poisson power series, 393 Poisson-stopped sums, , 213, 215 Poisson truncated normal, 370, 398, 483 rectangular mixture of Poissons, 368 stuttering Poisson, 195, 393 truncated-gamma mixed Poissons, 369 truncated-pearson Type III mixed Poissons, 370 Poisson regression model, 227, Pólya Aeppli distribution, 93, 243, 367, estimation, generalized Pólya Aeppli, genesis, 410 moment and other properties, probabilities, Polynomials Bell, 384, 461 Bernoulli, Charlier, 34 Chebyshev, 23, 34 Euler, 31 Gegenbauer, 500 generalized Laguerre, 33 Hermite, 25, Jacobi, 23, 34 Krawtchouk, 34

13 INDEX 645 Legendre, 22 orthogonal, Power parameter, 75 Power series distributions (PSD), characterizations, 77 estimation, properties, Probability axioms, 38 Probability of combined events, 38, Probability density function (pdf), 44 Probability mass function (pmf), 43 Probability measure, 38 Probability space, 38 Problème de rencontre (problem of coincidences), 434 Problem of points, 212 Propagation of error, 65 Pseudo-binomial variables, q-binomial theorem, 36 q-series (basic hypergeometric series), 34 37, , 464 reversed q-series, 464 q-series (basic hypergeometric) distributions, , absorption distribution, Bernoulli convolutions, bilateral q-series distributions, Euler, 205, 325, , 475, 477 generalized Euler, 472 Heine, 205, 466, 468, , 475 inverse absorption distribution, nonterminating distributions, q-binomial distributions, q-confluent hypergeometric, q-logarithmic, I, II, 474, 477 q-series related distributions, terminating distributions, Queueing theory distributions, 109, 187, 192, 202, 212, 215, 504, Randomly splitting a pack, 438 Random mappings, 336 Random sums, 361 Random variable (rv) continuous/discrete, difference/sum, 58, 60 Range, 61 Rao Rubin theorem, , 387 Records, Reed Frost chain binomial model, 105, Regression function, Risk function, 344 Runs, 430, applications, 453 distributions, runs up and down, success runs, Safety campaign model, 525 Sample mean/moments, Sampling from finite/infinite population, 109 inspection sampling, 381, mark release (capture recapture), 136, 188, 279, 526 Poissonian binomial sampling, Poisson trials model, quality control, 186, 279 size-biased, 150 snowball, 105 with additional replacements, 213, 255, with additional withdrawals, , 302 with replacement, 109 without replacement, 251, , 272 Sampling distribution, 55 Sampling schemes binomial, 109 Coolidge, hypergeometric, 251, 255 Lexian, 148 Poissonian binomial, Poisson Lexis, 148 Pólya, Scedasticity, 57 Series function, 75 Sesquimodal, 51 Shannon entropy, 48 49, 531 Size-bias, , 308 Skewness, 51 Stable distribution, 58 Statistic, 67 Statistical physics, 430, , 470 Stochastic processes Arfwedson, 213 Bernoulli damage, birth, death, immigration, emigration, 96, 99, 109, , 172, 185, 204, 215, 232, 261, , 513, 522 branching processes, 382, , 519 compound Poisson process, 393 damage process, Foster process, 215 mixed Poisson processes, 369 nonhomogeneous process, 95, 159, 188, 215 Poisson process, 95, 185, 187

14 646 INDEX Stochastic processes (Continued) Pólya process, 215, 304 renewal processes, 185, 512 time homogeneous, 95, 261 Yule Furry process, 215, 304 Stopped sum distributions, , 243, Gurland s theorem, 364, 383 Khatri and Patel s Types A, B, and C, Lévy s existence theorem, moments, multiple Poisson distributions, 385, , notation and terminology, Stuttering distributions stuttering negative binomial, 400 stuttering Poisson, 195, 393 Superimposition of distributions, 344 Support of a distribution, 43 Survival function, 45 Symbol ascending (rising) factorial, 2 binomial coefficient, 2 ceiling, 5 descending (falling) factorial, 2 floor, 5 multinomial coefficient, 4 Pochhammer s, 2 q-binomial coefficient, 35 signum, 5 Stirling numbers, 12 Symbolic calculus, 10 Symbolic representation of mixture, 345, Symbolic representation of stopped-sum, Tails relationships beta-binomial and beta-negative binomial, binomial and beta (F ), 78, 113 binomial and negative binomial, 221 hypergeometric and beta-negative binomial, hypergeometric and negative hypergeometric, negative binomial and beta, 78, 212 Poisson and gamma (chi-square), 78, 165, Taylor s power law, 494 Theorem of total probability, 41 Transformations arc sine (binomial data), 124 arc sinh (negative binomial data), 220 normalizing, 65 66, 124, 220 square root (Poisson data), 168 variance-stabilizing, 65, 124, 220 True contagion, 382 Truncation, 62 63, , , 211, , Two-crossings theorem, 363 Unidentifiable mixture, 349 Unimodality, 51 discrete α-unimodality, 51 Urn models, 136, 251, 255, 442, Consul, 526 Friedman, 525 leaking, 442, 446 Naor, play-the-winner, 526 Pólya, 209, 213, , Rutherford, stochastic replacements, Woodbury, 525 Vandermonde s theorem, 3, 21 Variance of a function, Variance-stabilizing transformation, 65, 124, 168, 220 Variance-to-mean relationship, 163, 216, 237 Visibility bias, 435 Wald Wolfowitz two-sample test, 453 Weighted distributions, , 240 Whitworth s theorem, 434 Wilcoxon Mann Whitney test, Wilks selection problem, 507 Zero-modified distributions, 194, 353, 356 hurdle distributions, Zero-truncated distributions, 137, , Zipf Estoup law, 527