About this title: Lagrangian expansions can be used to obtain numerous useful probability models, which have been applied to real life situations including, but not limited to: branching processes, queuing processes, stochastic processes, environmental toxicology, diffusion of information, ecology, strikes in industries, sales of new products, and production targets for optimum profits. This book presents a comprehensive, systematic treatment of the class of Lagrangian probability distributions, along with some of its families, their properties, and important applications.
Key features:
* Fills a gap in book literature
* Examines many new Lagrangian probability distributions, their numerous families, general and specific properties, and applications to a variety of different fields
* Presents background mathematical and statistical formulas for easy reference
* Detailed bibliography and index
* Exercises in many chapters
Graduate students and researchers with a good knowledge of standard statistical techniques and an interest in Lagrangian probability distributions will find this work valuable. It may be used as a reference text or in courses and seminars on Distribution Theory and Lagrangian Distributions. Applied scientists and researchers in environmental statistics, reliability, sales management, epidemiology, operations research, optimization in manufacturing and marketing, and infectious disease control will benefit immensely from the various applications in the book

Table Of Contents

Foreword

Preface

                           List of Tables

Abbreviations

             Preliminary Information

                    Introduction

                    Mathematical Symbols and Results

                           Combinatorial and Factorial Symbols

                           Gamma and Beta Functions

                           Difference and Differential Calculus

                           Stirling Numbers

                           Hypergeometric Functions

                           Lagrange Expansions

                           Abel and Gould Series

                           Faa di Bruno's Formula

                    Probabilistic and Statistical Results

                           Probabilities and Random Variables

                           Expected Values

                           Moments and Moment Generating Functions

                           Cumulants and Cumulant Generating Functions

                           Probability Generating Functions

                           Inference

             Lagrangian Probability Distributions

                    Introduction

                    Lagrangian Probability Distributions

                           Equivalence of the Two Classes of Lagrangian Distributions

                           Moments of Lagrangian Distributions

                           Applications of the Results on Mean and Variance

                           Convolution Property for Lagrangian Distributions

                           Probabilistic Structure of Lagrangian Distributions L(f; g; x)

                    Modified Power Series Distributions

                           Modified Power Series Based on Lagrange Expansions

                           MPSD as a Subclass of Lagrangian Distributions L(f; g; x)

                           Mean and Variance of a MPSD

                           Maximum Entropy Characterization of some MPSDs

                    Exercises

             Properties of General Lagrangian Distributions

                    Introduction

                    Central Moments of Lagrangian Distribution L(f; g; x)

                    Central Moments of Lagrangian Distribution L1(f1; g; y)

                           Cumulants of Lagrangian Distribution L1(f1; g; y)

                           Applications

                    Relations between the Two Classes L(f; g; x) and L1(f1; g; y)

                    Some Limit Theorems for Lagrangian Distributions

                    Exercises

             Quasi-Probability Models

                    Introduction

                    Quasi-Binomial Distribution I (QBD-I)

                           QBD-I as a True Probability Distribution

                           Mean and Variance of QBD-I

                           Negative Moments of QBD-I

                           QBD-I Model Based on Difference-Differential Equations

                           Maximum Likelihood Estimation

                    Quasi-Hypergeometric Distribution I

                    Quasi-Polya Distribution I

                    Quasi-Binomial Distribution II

                           QBD-II as a True Probability Model

                           Mean and Variance of QBD-II

                           Some Other Properties of QBD-II

                    Quasi-Hypergeometric Distribution II

                    Quasi-Polya Distribution II (QPD-II)

                           Special and Limiting Cases

                           Mean and Variance of QPD-II

                           Estimation of Parameters of QPD-II

                    Gould Series Distributions

                    Abel Series Distributions

                    Exercises

             Some Urn Models

                    Introduction

                    A Generalized Stochastic Urn Model

                           Some Interrelations among Probabilities

                           Recurrence Relation for Moments

                           Some Applications of Prem Model

                    Urn Model with Predetermined Strategy for Quasi-Binomial Distribution I

                           Sampling without Replacement from Urn B

                           Polya-type Sampling from Urn B

                    Urn Model with Predetermined Strategy for Quasi-Polya Distribution II

                           Sampling with Replacement from Urn D

                           Sampling without Replacement from Urn D

                           Urn Model with Inverse Sampling

                    Exercises

             Development of Models and Applications

                    Introduction

                    Branching Process

                    Queuing Process

                           G|D|1 Queue

                           M|G|1 Queue

                    Stochastic Model of Epidemics

                    Enumeration of Trees

                    Cascade Process

                    Exercises

             Modified Power Series Distributions

                    Introduction

                    Generating Functions

                    Moments, Cumulants, and Recurrence Relations

                    Other Interesting Properties

                    Estimation

                           Maximum Likelihood Estimation of θ

                           Minimum Variance Unbiased Estimation

                           Interval Estimation

                    Some Characterizations

                    Related Distributions

                           Inflated MPSD

                           Left Truncated MPSD

                    Exercises

             Some Basic Lagrangian Distributions

                    Introduction

                    Geeta Distribution

                           Definition

                           Generating Functions

                           Moments and Recurrence Relations

                           Other Interesting Properties

                           Physical Models Leading to Geeta Distribution

                           Estimation

                           Some Applications

                    Consul Distribution

                           Definition

                           Generating Functions

                           Moments and Recurrence Relations

                           Other Interesting Properties

                           Estimation

                           Some Applications

                    Borel Distribution

                           Definition

                           Generating Functions

                           Moments and Recurrence Relations

                           Other Interesting Properties

                           Estimation

                    Weighted Basic Lagrangian Distributions

                    Exercises

             Generalized Poisson Distribution

                    Introduction and Definition

                    Generating Functions

                    Moments, Cumulants, and Recurrence Relations

                    Physical Models Leading to GPD

                    Other Interesting Properties

                    Estimation

                           Point Estimation

                           Interval Estimation

                           Confidence Regions

                    Statistical Testing

                           Test about Parameters

                           Chi-Square Test

                           Empirical Distribution Function Test

                    Characterizations

                    Applications

                    Truncated Generalized Poisson Distribution

                    Restricted Generalized Poisson Distribution

                           Introduction and Definition

                           Estimation

                           Hypothesis Testing

                    Other Related Distributions

                           Compound and Weighted GPD

                           Differences of Two GP Variates

                           Absolute Difference of Two GP Variates

                           Distribution of Order Statistics when Sample Size Is a GP Variate

                           The Normal and Inverse Gaussian Distributions

                    Exercises

             Generalized Negative Binomial Distribution

                    Introduction and Definition

                    Generating Functions

                    Moments, Cumulants, and Recurrence Relations

                    Physical Models Leading to GNBD

                    Other Interesting Properties

                    Estimation

                           Point Estimation

                           Interval Estimation

                    Statistical Testing

                    Characterizations

                    Applications

                    Truncated Generalized Negative Binomial Distribution

                    Other Related Distributions

                           Poisson-Type Approximation

                           Generalized Logarithmic Series Distribution-Type Limit

                           Differences of Two GNB Variates

                           Weighted Generalized Negative Binomial Distribution

                    Exercises

             Generalized Logarithmic Series Distribution

                    Introduction and Definition

                    Generating Functions

                    Moments, Cumulants, and Recurrence Relations

                    Other Interesting Properties

                    Estimation

                           Point Estimation

                           Interval Estimation

                    Statistical Testing

                    Characterizations

                    Applications

                    Related Distributions

                    Exercises

             Lagrangian Katz Distribution

                    Introduction and Definition

                    Generating Functions

                    Moments, Cumulants, and Recurrence Relations

                    Other Important Properties

                    Estimation

                    Applications

                    Related Distributions

                           Basic LKD of Type I

                           Basic LKD of Type II

                           Basic GLKD of Type II

                    Exercises

             Random Walks and Jump Models

                    Introduction

                    Simplest Random Walk with Absorbing Barrier at the Origin

                    Gambler's Ruin Random Walk

                    Generating Function of Ruin Probabilities in a Polynomial Random Walk

                    Trinomial Random Walks

                    Quadrinomial Random Walks

                    Binomial Random Walk (Jumps) Model

                    Polynomial Random Jumps Model

                    General Random Jumps Model

                    Applications

                    Exercises

             Bivariate Lagrangian Distributions

                    Definitions and Generating Functions

                    Cumulants of Bivariate Lagrangian Distributions

                    Bivariate Modified Power Series Distributions

                           Introduction

                           Moments of BMPSD

                           Properties of BMPSD

                           Estimation of BMPSD

                    Some Bivariate Lagrangian Delta Distributions

                    Bivariate Lagrangian Poisson Distribution

                           Introduction

                           Moments and Properties

                           Special BLPD

                    Other Bivariate Lagrangian Distributions

                           Bivariate Lagrangian Binomial Distribution

                           Bivariate Lagrangian Negative Binomial Distribution

                           Bivariate Lagrangian Logarithmic Series Distribution

                           Bivariate Lagrangian Borel-Tanner Distribution

                           Bivariate Inverse Trinomial Distribution

                           Bivariate Quasi-Binomial Distribution

                    Exercises

             Multivariate Lagrangian Distributions

                    Introduction

                    Notation and Multivariate Lagrangian Distributions

                    Means and Variance-Covariance

                    Multivariate Lagrangian Distributions (Special Form)

                           Multivariate Lagrangian Poisson Distribution

                           Multivariate Lagrangian Negative Binomial Distribution

                           Multivariate Lagrangian Logarithmic Series Distribution

                           Multivariate Lagrangian Delta Distributions

                    Multivariate Modified Power Series Distributions

                           Multivariate Lagrangian Poisson Distribution

                           Multivariate Lagrangian Negative Binomial Distribution

                           Multivariate Lagrangian Logarithmic Series Distribution

                           Moments of the General Multivariate MPSD

                           Moments of Multivariate Lagrangian Poisson Distribution

                           Moments of Multivariate Lagrangian Negative Binomial Distribution

                    Multivariate MPSDs in Another Form

                    Multivariate Lagrangian Quasi-Polya Distribution

                    Applications of Multivariate Lagrangian Distributions

                           Queuing Processes

                           Random Branching Processes with k Types of Females

                    Exercises

             Computer Generation of Lagrangian Variables

                    Introduction and Generation Procedures

                           Inversion Method

                           Alias Method

                    Basic Lagrangian Random Variables

                    Simple Delta Lagrangian Random Variables

                    Generalized Poisson Random Variables

                    Generalized Negative Binomial Random Variables

                    Generalized Logarithmic Series Random Variables

                    Some Quasi-Type Random Variables

References

Index