Preface xvii 1 Preliminary Information 1 1.1 Mathematical Preliminaries, 1 1.1.1 Factorial and Combinatorial Conventions, 1 1.1.2 Gamma and Beta Functions, 5 1.1.3 Finite Difference Calculus, 10 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 1.1.7 Confluent Hypergeometric Functions (Kummer''s Functions), 23 1.

1.8 Generalized Hypergeometric Functions, 26 1.1.9 Bernoulli and Euler Numbers and Polynomials, 29 1.1.10 Integral Transforms, 32 1.1.11 Orthogonal Polynomials, 32 1.

1.12 Basic Hypergeometric Series, 34 1.2 Probability and Statistical Preliminaries, 37 1.2.1 Calculus of Probabilities, 37 1.2.2 Bayes''s Theorem, 41 1.2.

3 Random Variables, 43 1.2.4 Survival Concepts, 45 1.2.5 Expected Values, 47 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 1.2.10 Characteristic Functions, 57 1.2.

11 Probability Generating Functions, 58 1.2.12 Order Statistics, 61 1.2.13 Truncation and Censoring, 62 1.2.14 Mixture Distributions, 64 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 2 Families of Discrete Distributions 74 2.1 Lattice Distributions, 74 2.

2 Power Series Distributions, 75 2.2.1 Generalized Power Series Distributions, 75 2.2.2 Modified Power Series Distributions, 79 2.3 Difference-Equation Systems, 82 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 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 2.7 Factorial Series Distributions, 103 2.8 Distributions of Order-k, 105 2.9 q-Series Distributions, 106 3 Binomial Distribution 108 3.1 Definition, 108 3.2 Historical Remarks and Genesis, 109 3.

3 Moments, 109 3.4 Properties, 112 3.5 Order Statistics, 116 3.6 Approximations, Bounds, and Transformations, 116 3.6.1 Approximations, 116 3.6.2 Bounds, 122 3.

6.3 Transformations, 123 3.7 Computation, Tables, and Computer Generation, 124 3.7.1 Computation and Tables, 124 3.7.2 Computer Generation, 125 3.8 Estimation, 126 3.

8.1 Model Selection, 126 3.8.2 Point Estimation, 126 3.8.3 Confidence Intervals, 130 3.8.4 Model Verification, 133 3.

9 Characterizations, 134 3.10 Applications, 135 3.11 Truncated Binomial Distributions, 137 3.12 Other Related Distributions, 140 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 3.12.5 Chain Binomial Models, 151 3.12.6 Correlated Binomial Variables, 151 4 Poisson Distribution 156 4.

1 Definition, 156 4.2 Historical Remarks and Genesis, 156 4.2.1 Genesis, 156 4.2.2 Poissonian Approximations, 160 4.3 Moments, 161 4.4 Properties, 163 4.

5 Approximations, Bounds, and Transformations, 167 4.6 Computation, Tables, and Computer Generation, 170 4.6.1 Computation and Tables, 170 4.6.2 Computer Generation, 171 4.7 Estimation, 173 4.7.

1 Model Selection, 173 4.7.2 Point Estimation, 174 4.7.3 Confidence Intervals, 176 4.7.4 Model Verification, 178 4.8 Characterizations, 179 4.

9 Applications, 186 4.10 Truncated and Misrecorded Poisson Distributions, 188 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 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 4.12.5 Grouped Poisson Distributions, 202 4.12.6 Heine and Euler Distributions, 205 4.12.7 Intervened Poisson Distributions, 205 5 Negative Binomial Distribution 208 5.

1 Definition, 208 5.2 Geometric Distribution, 210 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 5.7 Computation and Tables, 220 5.8 Estimation, 222 5.

8.1 Model Selection, 222 5.8.2 P Unknown, 222 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 5.9.1 Geometric Distribution, 228 5.9.2 Negative Binomial Distribution, 231 5.10 Applications, 232 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 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 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 6 Hypergeometric Distributions 251 6.1 Definition, 251 6.2 Historical Remarks and Genesis, 252 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 6.2.4 Pólya Distributions, 258 6.2.5 Hypergeometric Distributions in General, 259 6.

3 Moments, 262 6.4 Properties, 265 6.5 Approximations and Bounds, 268 6.6 Tables, Computation, and Computer Generation, 271 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 6.8 Characterizations, 277 6.9 Applications, 279 6.9.1 Classical Hypergeometric Distribution, 279 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 6.

10.3 Yule Distribution, 287 6.10.4 Waring Distribution, 289 6.10.5 Narayana Distribution, 291 6.11 Related Distributions, 293 6.11.

1 Extended Hypergeometric Distributions, 293 6.11.2 Generalized Hypergeometric Probability Distributions, 296 6.11.3 Generalized Hypergeometric Factorial Moment Distributions, 298 6.11.4 Other Related Distributions, 299 7 Logarithmic and Lagrangian Distributions 302 7.1 Logarithmic Distribution, 302 7.

1.1 Definition, 302 7.1.2 Historical Remarks and Genesis, 303 7.1.3 Moments, 305 7.1.4 Properties, 307 7.

1.5 Approximations and Bounds, 309 7.1.6 Computation, Tables, and Computer Generation, 310 7.1.7 Estimation, 311 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 7.1.12 Other Related Distributions, 321 7.

2 Lagrangian Distributions, 325 7.2.1 Otter''s Multiplicative Process, 326 7.2.2 Borel Distribution, 328 7.2.3 Consul Distribution, 329 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 7.2.

8 Lagrangian Logarithmic Distribution, 341 7.2.9 Lagrangian Distributions of the Second Kind, 342 8 Mixture Distributions 343 8.1 Basic Ideas, 343 8.1.1 Introduction, 343 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 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 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 8.

3.3 Examples of Poisson Mixtures, 365 8.3.4 Mixtures of Binomial Distributions, 373 8.3.5 Examples.