Lagrangian Probability Distributions

About the Authors......Page 2
Foreword......Page 7
Preface......Page 9
Contents......Page 11
List of Tables......Page 18
Abbreviations......Page 19
Abbreviations Continued......Page 20
1.2.1 Combinatorial and Factorial Symbols......Page 22
1.2.2 Gamma and Beta Functions......Page 25
1.2.3 Difference and Differential Calculus......Page 26
1.2.4 Stirling Numbers......Page 28
1.2.5 Hypergeometric Functions......Page 30
1.2.6 Lagrange Expansions......Page 31
1.2.8 Fa`a di Bruno s Formula......Page 34
1.3.1 Probabilities and Random Variables......Page 35
1.3.2 Expected Values......Page 36
1.3.3 Moments and Moment Generating Functions......Page 37
1.3.5 Probability Generating Functions......Page 39
1.3.6 Inference......Page 40
2.1 Introduction......Page 42
Basic Lagrangian Distributions......Page 43
Delta Lagrangian Distributions......Page 44
General Lagrangian Distributions......Page 46
2.2.1 Equivalence of the Two Classes of Lagrangian Distributions......Page 51
2.2.2 Moments of Lagrangian Distributions......Page 54
2.2.4 Convolution Property for Lagrangian Distributions......Page 57
2.2.5 Probabilistic Structure of Lagrangian Distributions......Page 60
2.3 Modi“ed Power Series Distributions......Page 62
2.3.2 MPSD as a Subclass of Lagrangian Distributions......Page 63
2.3.3 Mean and Variance of a MPSD......Page 66
2.3.4 Maximum Entropy Characterization of some MPSDs......Page 67
2.4 Exercises......Page 69
3.2 Central Moments of Lagrangian Distribution......Page 72
Cumulants of Lagrangian Distributions L( f ; g; x)......Page 74
3.3 Central Moments of Lagrangian Distribution......Page 77
3.3.1 Cumulants of Lagrangian Distribution......Page 81
3.3.2 Applications......Page 82
3.4 Relations between the Two Classes L( f ; g; x) and L1( f1; g; y)......Page 83
Some Examples......Page 85
3.5 Some Limit Theorems for Lagrangian Distributions......Page 87
3.6 Exercises......Page 88
4.1 Introduction......Page 90
4.2.1 QBD-I as a True Probability Distribution......Page 91
4.2.2 Mean and Variance of QBD-I......Page 92
4.2.3 Negative Moments of QBD-I......Page 94
4.2.4 QBD-I Model Based on Difference-Differential Equations......Page 96
4.2.5 Maximum Likelihood Estimation......Page 98
4.3 Quasi-Hypergeometric Distribution I......Page 101
4.4 Quasi-P´olya Distribution I......Page 102
4.5 Quasi-Binomial Distribution II......Page 103
4.5.2 Mean and Variance of QBD-II......Page 104
4.6 Quasi-Hypergeometric Distribution II......Page 106
4.7 Quasi-P´olya Distribution II (QPD-II)......Page 107
4.7.1 Special and Limiting Cases......Page 108
4.7.3 Estimation of Parameters of QPD-II......Page 109
4.8 Gould Series Distributions......Page 110
4.10 Exercises......Page 111
5.1 Introduction......Page 114
5.2 A Generalized Stochastic Urn Model......Page 115
5.2.1 Some Interrelations among Probabilities......Page 121
5.2.2 Recurrence Relation for Moments......Page 122
5.2.3 Some Applications of Prem Model......Page 124
5.3.1 Sampling without Replacement from Urn B......Page 125
5.4 Urn Model with Predetermined Strategy for Quasi-P´olya Distribution II......Page 126
5.4.1 Sampling with Replacement from Urn D......Page 127
5.4.3 Urn Model with Inverse Sampling......Page 128
5.5 Exercises......Page 129
6.2 Branching Process......Page 130
6.3 Queuing Process......Page 132
6.3.1 G|D|1 Queue......Page 133
6.3.2 M|G|1 Queue......Page 134
6.4 Stochastic Model of Epidemics......Page 136
6.5 Enumeration of Trees......Page 137
6.6 Cascade Process......Page 138
6.7 Exercises......Page 139
7.1 Introduction......Page 142
7.3 Moments, Cumulants, and Recurrence Relations......Page 143
Negative and Incomplete Moments......Page 145
7.4 Other Interesting Properties......Page 146
7.5.1 Maximum Likelihood Estimation of......Page 149
7.5.2 Minimum Variance Unbiased Estimation......Page 151
7.6 Some Characterizations......Page 153
7.7.1 In”ated MPSD......Page 157
7.7.2 Left Truncated MPSD......Page 158
7.8 Exercises......Page 161
8.2.1 De“nition......Page 164
8.2.2 Generating Functions......Page 165
8.2.4 Other Interesting Properties......Page 166
8.2.5 Physical Models Leading to Geeta Distribution......Page 167
8.2.6 Estimation......Page 169
8.2.7 Some Applications......Page 171
8.3.1 De“nition......Page 172
8.3.3 Moments and Recurrence Relations......Page 173
8.3.4 Other Interesting Properties......Page 174
8.3.5 Estimation......Page 175
8.3.6 Some Applications......Page 178
8.4.2 Generating Functions......Page 179
8.4.4 Other Interesting Properties......Page 180
8.4.5 Estimation......Page 181
8.5 Weighted Basic Lagrangian Distributions......Page 182
8.6 Exercises......Page 183
9.1 Introduction and De“nition......Page 186
9.2 Generating Functions......Page 187
9.4 Physical Models Leading to GPD......Page 188
Models Based on Difference-Differential Equations......Page 189
Thermodynamic Process......Page 190
9.6.1 Point Estimation Moment Estimation......Page 191
Maximum Likelihood Estimation......Page 192
Empirical Weighted Rates of Change Estimation......Page 193
9.6.2 Interval Estimation......Page 194
9.7.1 Test about Parameters......Page 196
9.7.2 Chi-Square Test......Page 197
9.8 Characterizations......Page 198
9.9 Applications......Page 200
9.10 Truncated Generalized Poisson Distribution......Page 201
9.11.2 Estimation......Page 203
9.11.3 Hypothesis Testing......Page 206
9.12.1 Compound and Weighted GPD......Page 207
9.12.2 Differences of Two GP Variates......Page 208
9.12.4 Distribution of Order Statistics when Sample Size Is a GP Variate......Page 209
9.13 Exercises......Page 210
10.1 Introduction and De“nition......Page 212
10.3 Moments, Cumulants, and Recurrence Relations......Page 213
RandomWalkModel......Page 215
Mixture Distribution......Page 216
Urn Model......Page 217
Unimodality......Page 218
Relations between Probabilities......Page 220
10.6 Estimation......Page 221
Estimation Based on Moments and Zero-Class Frequency......Page 222
Maximum Likelihood Estimation......Page 223
MVU Estimation when β and m Are Known......Page 224
10.6.2 Interval Estimation......Page 225
10.7 Statistical Testing......Page 226
10.8 Characterizations......Page 228
Queuing Process......Page 236
Molecular Weight Distribution in Polymers......Page 237
10.10 Truncated Generalized Negative Binomial Distribution......Page 238
10.11.2 Generalized Logarithmic Series Distribution-Type Limit......Page 239
10.11.3 Differences of Two GNB Variates......Page 240
10.12 Exercises......Page 241
11.1 Introduction and De“nition......Page 244
11.2 Generating Functions......Page 245
11.3 Moments, Cumulants, and Recurrence Relations......Page 246
11.4 Other Interesting Properties......Page 247
Moment Estimates......Page 250
Proportion of Ones and Sample Mean Method......Page 251
Minimum Variance Unbiased Estimation......Page 252
11.6 Statistical Testing......Page 254
11.7 Characterizations......Page 255
Truncated GLSD......Page 257
Modi“ed GLSD......Page 258
11.10 Exercises......Page 259
12.2 Generating Functions......Page 262
12.3 Moments, Cumulants, and Recurrence Relations......Page 263
12.4 Other Important Properties......Page 265
12.5 Estimation......Page 266
Estimation Based on Moments and Zero-Class Frequency......Page 267
Maximum Likelihood Estimation......Page 268
12.7.1 Basic LKD of Type I......Page 269
12.7.2 Basic LKD of Type II......Page 270
12.7.3 Basic GLKD of Type II......Page 271
12.8 Exercises......Page 272
13.1 Introduction......Page 274
13.3 Gambler s Ruin Random Walk......Page 275
13.4 Generating Function of Ruin Probabilities in a Polynomial Random Walk......Page 276
Probability of Capital i after the Nth Step......Page 279
13.5 Trinomial Random Walks......Page 280
13.6 Quadrinomial Random Walks......Page 281
13.7 Binomial Random Walk (Jumps) Model......Page 283
13.8 Polynomial Random Jumps Model......Page 284
13.9 General Random Jumps Model......Page 285
13.10 Applications......Page 286
13.11 Exercises......Page 287
14.1 De“nitions and Generating Functions......Page 290
14.2 Cumulants of Bivariate Lagrangian Distributions......Page 292
14.3.1 Introduction......Page 293
14.3.2 Moments of BMPSD......Page 294
14.3.3 Properties of BMPSD......Page 296
14.3.4 Estimation of BMPSD Maximum Likelihood Estimation......Page 297
MVU Estimation......Page 298
Bivariate Modi“ed Double Poisson Model......Page 299
Bivariate Modi“ed Delta Binomial Model......Page 300
14.4 Some Bivariate Lagrangian Delta Distributions......Page 301
14.5.1 Introduction......Page 302
14.5.2 Moments and Properties......Page 303
14.6.1 Bivariate Lagrangian Binomial Distribution......Page 304
14.6.2 Bivariate Lagrangian Negative Binomial Distribution......Page 305
14.6.3 Bivariate Lagrangian Logarithmic Series Distribution......Page 307
14.6.4 Bivariate Lagrangian Borel…Tanner Distribution......Page 308
14.6.5 Bivariate Inverse Trinomial Distribution......Page 309
14.6.6 Bivariate Quasi-Binomial Distribution......Page 310
14.7 Exercises......Page 311
15.2 Notation and Multivariate Lagrangian Distributions......Page 314
15.3 Means and Variance-Covariance......Page 318
15.4.1 Multivariate Lagrangian Poisson Distribution......Page 320
15.4.3 Multivariate Lagrangian Logarithmic Series Distribution......Page 321
15.4.4 Multivariate Lagrangian Delta Distributions......Page 322
15.5 Multivariate Modi“ed Power Series Distributions......Page 323
15.5.1 Multivariate Lagrangian Poisson Distribution......Page 324
15.5.4 Moments of the General Multivariate MPSD......Page 325
15.5.5 Moments of Multivariate Lagrangian Poisson Distribution......Page 326
15.5.6 Moments of Multivariate Lagrangian Negative Binomial Distribution......Page 328
15.6 Multivariate MPSDs in Another Form......Page 329
15.8 Applications of Multivariate Lagrangian Distributions......Page 333
15.8.1 Queuing Processes......Page 334
15.8.2 Random Branching Processes with k Types of Females......Page 335
15.9 Exercises......Page 336
16.1.1 Inversion Method......Page 338
16.1.2 Alias Method......Page 339
16.2 Basic Lagrangian Random Variables......Page 340
16.3 Simple Delta Lagrangian Random Variables......Page 342
16.4 Generalized Poisson Random Variables......Page 345
16.5 Generalized Negative Binomial Random Variables......Page 347
16.6 Generalized Logarithmic Series Random Variables......Page 351
16.7 Some Quasi-Type Random Variables......Page 355
References......Page 358
Index......Page 368