Univariate Discrete Distributions

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). 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.

• Preface xvii1 Preliminary Information 11.1 Mathematical Preliminaries 11.1.1 Factorial and Combinatorial Conventions 11.1.2 Gamma and Beta Functions 51.1.3 Finite Difference Calculus 101.1.4 Differential Calculus 141.1.5 Incomplete Gamma and Beta Functions and Other Gamma-Related Functions 161.1.6 Gaussian Hypergeometric Functions 201.1.7 Confluent Hypergeometric Functions (Kummer’s Functions) 231.1.8 Generalized Hypergeometric Functions 261.1.9 Bernoulli and Euler Numbers and Polynomials 291.1.10 Integral Transforms 321.1.11 Orthogonal Polynomials 321.1.12 Basic Hypergeometric Series 341.2 Probability and Statistical Preliminaries 371.2.1 Calculus of Probabilities 371.2.2 Bayes’s Theorem 411.2.3 Random Variables 431.2.4 Survival Concepts 451.2.5 Expected Values 471.2.6 Inequalities 491.2.7 Moments and Moment Generating Functions 501.2.8 Cumulants and Cumulant Generating Functions 541.2.9 Joint Moments and Cumulants 561.2.10 Characteristic Functions 571.2.11 Probability Generating Functions 581.2.12 Order Statistics 611.2.13 Truncation and Censoring 621.2.14 Mixture Distributions 641.2.15 Variance of a Function 651.2.16 Estimation 661.2.17 General Comments on the Computer Generation of Discrete Random Variables 711.2.18 Computer Software 732 Families of Discrete Distributions 742.1 Lattice Distributions 742.2 Power Series Distributions 752.2.1 Generalized Power Series Distributions 752.2.2 Modified Power Series Distributions 792.3 Difference-Equation Systems 822.3.1 Katz and Extended Katz Families 822.3.2 Sundt and Jewell Family 852.3.3 Ord’s Family 872.4 Kemp Families 892.4.1 Generalized Hypergeometric Probability Distributions 892.4.2 Generalized Hypergeometric Factorial Moment Distributions 962.5 Distributions Based on Lagrangian Expansions 992.6 Gould and Abel Distributions 1012.7 Factorial Series Distributions 1032.8 Distributions of Order-k 1052.9 q-Series Distributions 1063 Binomial Distribution 1083.1 Definition 1083.2 Historical Remarks and Genesis 1093.3 Moments 1093.4 Properties 1123.5 Order Statistics 1163.6 Approximations, Bounds, and Transformations 1163.6.1 Approximations 1163.6.2 Bounds 1223.6.3 Transformations 1233.7 Computation, Tables, and Computer Generation 1243.7.1 Computation and Tables 1243.7.2 Computer Generation 1253.8 Estimation 1263.8.1 Model Selection 1263.8.2 Point Estimation 1263.8.3 Confidence Intervals 1303.8.4 Model Verification 1333.9 Characterizations 1343.10 Applications 1353.11 Truncated Binomial Distributions 1373.12 Other Related Distributions 1403.12.1 Limiting Forms 1403.12.2 Sums and Differences of Binomial-Type Variables 1403.12.3 Poissonian Binomial, Lexian, and Coolidge Schemes 1443.12.4 Weighted Binomial Distributions 1493.12.5 Chain Binomial Models 1513.12.6 Correlated Binomial Variables 1514 Poisson Distribution 1564.1 Definition 1564.2 Historical Remarks and Genesis 1564.2.1 Genesis 1564.2.2 Poissonian Approximations 1604.3 Moments 1614.4 Properties 1634.5 Approximations, Bounds, and Transformations 1674.6 Computation, Tables, and Computer Generation 1704.6.1 Computation and Tables 1704.6.2 Computer Generation 1714.7 Estimation 1734.7.1 Model Selection 1734.7.2 Point Estimation 1744.7.3 Confidence Intervals 1764.7.4 Model Verification 1784.8 Characterizations 1794.9 Applications 1864.10 Truncated and Misrecorded Poisson Distributions 1884.10.1 Left Truncation 1884.10.2 Right Truncation and Double Truncation 1914.10.3 Misrecorded Poisson Distributions 1934.11 Poisson–Stopped Sum Distributions 1954.12 Other Related Distributions 1964.12.1 Normal Distribution 1964.12.2 Gamma Distribution 1964.12.3 Sums and Differences of Poisson Variates 1974.12.4 Hyper-Poisson Distributions 1994.12.5 Grouped Poisson Distributions 2024.12.6 Heine and Euler Distributions 2054.12.7 Intervened Poisson Distributions 2055 Negative Binomial Distribution 2085.1 Definition 2085.2 Geometric Distribution 2105.3 Historical Remarks and Genesis of Negative Binomial Distribution 2125.4 Moments 2155.5 Properties 2175.6 Approximations and Transformations 2185.7 Computation and Tables 2205.8 Estimation 2225.8.1 Model Selection 2225.8.2 P Unknown 2225.8.3 Both Parameters Unknown 2235.8.4 Data Sets with a Common Parameter 2265.8.5 Recent Developments 2275.9 Characterizations 2285.9.1 Geometric Distribution 2285.9.2 Negative Binomial Distribution 2315.10 Applications 2325.11 Truncated Negative Binomial Distributions 2335.12 Related Distributions 2365.12.1 Limiting Forms 2365.12.2 Extended Negative Binomial Model 2375.12.3 Lagrangian Generalized Negative Binomial Distribution 2395.12.4 Weighted Negative Binomial Distributions 2405.12.5 Convolutions Involving Negative Binomial Variates 2415.12.6 Pascal–Poisson Distribution 2435.12.7 Minimum (Riff–Shuffle) and Maximum Negative Binomial Distributions 2445.12.8 Condensed Negative Binomial Distributions 2465.12.9 Other Related Distributions 2476 Hypergeometric Distributions 2516.1 Definition 2516.2 Historical Remarks and Genesis 2526.2.1 Classical Hypergeometric Distribution 2526.2.2 Beta–Binomial Distribution, Negative (Inverse) Hypergeometric Distribution: Hypergeometric Waiting-Time Distribution 2536.2.3 Beta–Negative Binomial Distribution: Beta–Pascal Distribution, Generalized Waring Distribution 2566.2.4 Pólya Distributions 2586.2.5 Hypergeometric Distributions in General 2596.3 Moments 2626.4 Properties 2656.5 Approximations and Bounds 2686.6 Tables Computation and Computer Generation 2716.7 Estimation 2726.7.1 Classical Hypergeometric Distribution 2736.7.2 Negative (Inverse) Hypergeometric Distribution: Beta–Binomial Distribution 2746.7.3 Beta–Pascal Distribution 2766.8 Characterizations 2776.9 Applications 2796.9.1 Classical Hypergeometric Distribution 2796.9.2 Negative (Inverse) Hypergeometric Distribution: Beta–Binomial Distribution 2816.9.3 Beta–Negative Binomial Distribution: Beta–Pascal Distribution, Generalized Waring Distribution 2836.10 Special Cases 2836.10.1 Discrete Rectangular Distribution 2836.10.2 Distribution of Leads in Coin Tossing 2866.10.3 Yule Distribution 2876.10.4 Waring Distribution 2896.10.5 Narayana Distribution 2916.11 Related Distributions 2936.11.1 Extended Hypergeometric Distributions 2936.11.2 Generalized Hypergeometric Probability Distributions 2966.11.3 Generalized Hypergeometric Factorial Moment Distributions 2986.11.4 Other Related Distributions 2997 Logarithmic and Lagrangian Distributions 3027.1 Logarithmic Distribution 3027.1.1 Definition 3027.1.2 Historical Remarks and Genesis 3037.1.3 Moments 3057.1.4 Properties 3077.1.5 Approximations and Bounds 3097.1.6 Computation, Tables, and Computer Generation 3107.1.7 Estimation 3117.1.8 Characterizations 3157.1.9 Applications 3167.1.10 Truncated and Modified Logarithmic Distributions 3177.1.11 Generalizations of the Logarithmic Distribution 3197.1.12 Other Related Distributions 3217.2 Lagrangian Distributions 3257.2.1 Otter’s Multiplicative Process 3267.2.2 Borel Distribution 3287.2.3 Consul Distribution 3297.2.4 Geeta Distribution 3307.2.5 General Lagrangian Distributions of the First Kind 3317.2.6 Lagrangian Poisson Distribution 3367.2.7 Lagrangian Negative Binomial Distribution 3407.2.8 Lagrangian Logarithmic Distribution 3417.2.9 Lagrangian Distributions of the Second Kind 3428 Mixture Distributions 3438.1 Basic Ideas 3438.1.1 Introduction 3438.1.2 Finite Mixtures 3448.1.3 Varying Parameters 3458.1.4 Bayesian Interpretation 3478.2 Finite Mixtures of Discrete Distributions 3478.2.1 Parameters of Finite Mixtures 3478.2.2 Parameter Estimation 3498.2.3 Zero-Modified and Hurdle Distributions 3518.2.4 Examples of Zero-Modified Distributions 3538.2.5 Finite Poisson Mixtures 3578.2.6 Finite Binomial Mixtures 3588.2.7 Other Finite Mixtures of Discrete Distributions 3598.3 Continuous and Countable Mixtures of Discrete Distributions 3608.3.1 Properties of General Mixed Distributions 3608.3.2 Properties of Mixed Poisson Distributions 3628.3.3 Examples of Poisson Mixtures 3658.3.4 Mixtures of Binomial Distributions 3738.3.5 Examples of Binomial Mixtures 3748.3.6 Other Continuous and Countable Mixtures of Discrete Distributions 3768.4 Gamma and Beta Mixing Distributions 3789 Stopped-Sum Distributions 3819.1 Generalized and Generalizing Distributions 3819.2 Damage Processes 3869.3 Poisson–Stopped Sum (Multiple Poisson) Distributions 3889.4 Hermite Distribution 3949.5 Poisson–Binomial Distribution 4009.6 Neyman Type A Distribution 4039.6.1 Definition 4039.6.2 Moment Properties 4059.6.3 Tables and Approximations 4069.6.4 Estimation 4079.6.5 Applications 4099.7 Pólya–Aeppli Distribution 4109.8 Generalized Pólya–Aeppli (Poisson–Negative Binomial) Distribution 4149.9 Generalizations of Neyman Type A Distribution 4169.10 Thomas Distribution 4219.11 Borel–Tanner Distribution: Lagrangian Poisson Distribution 4239.12 Other Poisson–Stopped Sum (multiple Poisson) Distributions 4259.13 Other Families of Stopped-Sum Distributions 42610 Matching, Occupancy, Runs, and q-Series Distributions 43010.1 Introduction 43010.2 Probabilities of Combined Events 43110.3 Matching Distributions 43410.4 Occupancy Distributions 43910.4.1 Classical Occupancy and Coupon Collecting 43910.4.2 Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac Statistics 44410.4.3 Specified Occupancy and Grassia–Binomial Distributions 44610.5 Record Value Distributions 44810.6 Runs Distributions 45010.6.1 Runs of Like Elements 45010.6.2 Runs Up and Down 45310.7 Distributions of Order k 45410.7.1 Early Work on Success Runs Distributions 45410.7.2 Geometric Distribution of Order k 45610.7.3 Negative Binomial Distributions of Order k 45810.7.4 Poisson and Logarithmic Distributions of Order k 45910.7.5 Binomial Distributions of Order k 46110.7.6 Further Distributions of Order k 46310.8 q-Series Distributions 46410.8.1 Terminating Distributions 46510.8.2 q-Series Distributions with Infinite Support 47010.8.3 Bilateral q-Series Distributions 47410.8.4 q-Series Related Distributions 47611 Parametric Regression Models and Miscellanea 47811.1 Parametric Regression Models 47811.1.1 Introduction 47811.1.2 Tweedie–Poisson Family 48011.1.3 Negative Binomial Regression Models 48211.1.4 Poisson Lognormal Model 48311.1.5 Poisson–Inverse Gaussian (Sichel) Model 48411.1.6 Poisson Polynomial Distribution 48711.1.7 Weighted Poisson Distributions 48811.1.8 Double-Poisson and Double-Binomial Distributions 48911.1.9 Simplex–Binomial Mixture Model 49011.2 Miscellaneous Discrete Distributions 49111.2.1 Dandekar’s Modified Binomial and Poisson Models 49111.2.2 Digamma and Trigamma Distributions 49211.2.3 Discrete Adès Distribution 49411.2.4 Discrete Bessel Distribution 49511.2.5 Discrete Mittag–Leffler Distribution 49611.2.6 Discrete Student’s t Distribution 49811.2.7 Feller–Arley and Gegenbauer Distributions 49911.2.8 Gram–Charlier Type B Distributions 50111.2.9 “Interrupted” Distributions 50211.2.10 Lost-Games Distributions 50311.2.11 Luria–Delbrück Distribution 50511.2.12 Naor’s Distribution 50711.2.13 Partial-Sums Distributions 50811.2.14 Queueing Theory Distributions 51211.2.15 Reliability and Survival Distributions 51411.2.16 Skellam–Haldane Gene Frequency Distribution 51911.2.17 Steyn’s Two-Parameter Power Series Distributions 52111.2.18 Univariate Multinomial-Type Distributions 52211.2.19 Urn Models with Stochastic Replacements 52411.2.20 Zipf-Related Distributions 52611.2.21 Haight’s Zeta Distributions 533Bibliography 535Abbreviations 631Index 633

“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 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) "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)