Univariate Discrete Distributions
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1 Univariate Discrete Distributions Second Edition NORMAN L. JOHNSON University of North Carolina Chapel Hill, North Carolina SAMUEL KOTZ University of Maryland College Park, Maryland ADRIENNE W. KEMP University of St Andrews St Andrews, Scotland A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Brisbane Toronto Singapore
2 Contents Preface List of Tables xv xix 1 Preliminary Information 1 Introduction, 1 A Mathematical Preliminaries, 1 Al Factorial and Combinatorial Conventions, 1 A2 Gamma and Beta Functions, 4 A3 Finite Difference Calculus, 8 A4 Differential Calculus, 12 A5 Incomplete Gamma and Beta Functions, and Other Gamma- Related Functions, 14 A6 Gaussian Hypergeometric Functions, 17 A7 Confluent Hypergeometric Functions (Kummer's Functions), 19 A8 Generalized Hypergeometric Functions, 22 A9 Bernoulli and Euler Numbers and Polynomials, 25 A10 Integral Transforms, 27 All Orthogonal Polynomials, 27 A12 Miscellaneous Topics, 29 В Probability and Statistical Preliminaries, 32 Bl Calculus of Probabilities, 32 B2 Real Variables, 35 B3 Bayes' Theorem, 38 B4 Expected Values, 39 B5 Moments and Moment Generating Functions, 40 B6 Cumulants and Cumulant Generating Functions, 44 vii
3 viii CONTENTS B7 Joint Moments and Cumulants, 46 B8 Characteristic Functions, 47 B9 Probability Generating Functions, 48 BIO Order Statistics, 51 Bll Truncation and Censoring, 52 B12 Mixture Distributions, 53 B13 Variance of a Function, 54 B14 Geometrical Concepts, 55 В15 Inference, 56 С Computer Generation of Univariate Discrete Random Variables, 61 CI General Comments, 61 C2 Distributionally Non-Specific Generation Procedures, 62 C3 Binomial Random Variables, 64 C4 Poisson Random Variables, 65 C5 Negative Binomial Random Variables, 66 C6 Hypergeometric Random Variables, 67 C7 Logarithmic Random Variables, 68 2 Families of Discrete Distributions 69 1 Lattice Distributions, 69 2 Power Series Distributions, Generalized Power Series Distributions, Modified Power Series Distributions, 74 3 Difference Equation Systems, Katz and Extended Katz Families, Ord's Family, 81 4 Kemp Families, Generalized Hypergeometric Probability Distributions, Generalized Hypergeometric Factorial Moment Distributions, 91 5 Distributions Based on Lagrangian Expansions, Otter's Multiplicative Process, Lagrangian Distributions, Gould and Abel Distributions, Factorial Series Distributions, Binomial Distribution Definition, Historical Remarks and Genesis, 106
4 CONTENTS IX 3 Moments, Properties, Order Statistics, Approximations, Bounds, and Transformations, Approximations, Bounds, Transformations, Computation and Tables, Estimation, Model Selection, Point Estimation, Confidence Intervals, Model Verification, Characterizations, Applications, Truncated Binomial Distributions, Other Related Distributions, Limiting Forms, Poissonian Binomial, Lexian, and Coolidge Schemes, Binomial-Binomial Lagrangian Distributions, Weighted Binomial Distributions, Pseudo-Binomial Variables, Correlated Binomial Variables, Poisson Distribution Definition, Historical Remarks and Genesis, Moments, Properties, Approximations, Bounds, and Transformations, Computation and Tables, Estimation, Model Selection, Point Estimation, Confidence Intervals, Model Verification, Characterizations, Applications, Truncated and Misrecorded Poisson Distributions, 181
5 X CONTENTS 10.1 Left-Truncation, Right-Truncation and Double-Truncation, Misrecorded Poisson Distributions, Poisson-Stopped-Sum Distributions, Other Related Distributions, The Normal Distribution, The Gamma Distribution, Sums and Differences of Poisson Variates, Hyper-Poisson Distributions, Grouped Poisson Distributions, Heine and Euler Distributions, Negative Binomial Distribution Definition, Geometric Distribution, Historical Remarks and Genesis, Moments, Properties, Approximations and Transformations, Computation and Tables, Estimation, Model Selection, P Unknown, Both Parameters Unknown, Data Sets with a Common Parameter, Characterizations, Geometric Distribution, Negative Binomial Distribution, Applications, Truncated Negative Binomial Distributions, Other Related Distributions, Limiting Forms, Engen's Extended Negative Binomial Model, Lagrangian "Generalized Negative Binomial Distribution", Weighted Negative Binomial Distributions, Convolutions Involving Negative Binomial Variates, Pascal-Poisson Distribution, A Riff-Shuffle Distribution, 234
6 Hypergeometric Distributions Definition, Historical Remarks and Genesis, Classical Hypergeometric Distribution, Negative (Inverse) Hypergeometric Distribution: Hypergeometric Waiting-Time Distribution, Beta-Binomial Distribution, Beta-Negative-Binomial Distribution: Beta-Pascal Distribution, Generalized Waring Distribution, Poly a Distributions: Generalized Hypergeometric Distributions, Moments, Properties, Approximations and Bounds, Tables and Computation, Estimation, Classical Hypergeometric Distribution, Negative (Inverse) Hypergeometric Distribution, Beta-Pascal Distribution, Characterizations, Applications, Classical Hypergeometric Distribution, Negative (Inverse) Hypergeometric Distribution: Beta-Binomial Distribution, Beta-Negative-Binomial Distribution: Beta-Pascal Distribution, Generalized Waring Distribution, Special Cases, Discrete Rectangular Distribution, Distribution of Leads in Coin Tossing, Yule Distribution, Waring Distribution, Extended Hypergeometric Distributions, Other Related Distributions, 282 Logarithmic Distribution Definition, Historical Remarks and Genesis, Moments, Properties, 290 XI
7 5 Approximations and Bounds, Computation and Tables, Estimation, Model Selection, Point Estimation and Confidence Intervals, Characterizations, Applications, Truncated and Modified Logarithmic Distributions, Other Related Distributions, 300 Mixture Distributions Introduction, Finite Mixtures of Discrete Distributions, Parameters of Finite Mixtures, Zero-Modified Distributions, Finite Poisson Mixtures, Finite Binomial Mixtures, Other Finite Mixtures of Discrete Distributions, Continuous and Countable Mixtures of Discrete Distributions, Three Important Theorems, Mixtures of Poisson Distributions, Mixtures of Binomial Distributions, Other Continuous and Countable Mixtures of Discrete Distributions, 337 Generalized (Stopped-Sum) Distributions Introduction, Damage Processes, Poisson-Stopped-Sum Distributions: Generalized Poisson Distributions, Hermite Distribution, Poisson-Binomial Distribution, Neyman Type A Distribution, Definition, Moment Properties, Tables and Approximations, Estimation, Applications, Polya-Aeppli Distribution, 378
8 CONTENTS xiii 8 Poisson-Pascal Distribution: Poisson-Negative Binomial Distribution, Generalized Pölya-Aeppli Distribution, Generalizations of the Neyman Type A Distribution, Thomas Distribution, Lagrangian Poisson Distribution: Shifted Borel-Tanner Distribution, Other Families of Stopped-Sum Distributions, Matching, Occupancy, and Runs Distributions Introduction, Probabilities of Combined Events, Matching Distributions, Occupancy Distributions, Classical Occupancy and Coupon Collecting, Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac Statistics, Runs Distributions, Runs of Like Elements, Runs Up and Down, Distributions of Order k, Success Runs Distributions, Philippou, Aki, and Hirano's Distributions of Order k, Further Distributions of Order k, Miscellaneous Discrete Distributions Absorption Distribution, Dandekar's Modified Binomial and Poisson Distributions, Digamma and Trigamma Distributions, Discrete Ades Distribution, Discrete Student's f-distribution, Geeta Distribution, Gegenbauer Distribution: Negative Binomial*Pseudo-Negative Binomial Convolution, Gram-Charlier Type В Distributions, "Interrupted" Distributions, Lost-Games Distributions, Naor's Distribution, Partial-Sums Distributions, Queueing Theory Distributions, 451
9 xiv CONTENTS 14 Record-Value Distributions, Sichel Distribution: Poisson-Inverse Gaussian Distribution, Skellam's Gene Frequency Distribution, Steyn's Two-Parameter Power Series Distributions, Univariate Multinomial-Type Distributions, Urn Models with Stochastic Replacements, Zipf and Zeta Distributions, 465 Bibliography 473 Abbreviations 549 Index 551
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