This is an introductory book on discrete statistical distributions and its applications. It discusses only those that are widely used in the applications of probability and statistics in everyday life. The purpose is to give a self-contained introduction to classical discrete distributions in statistics. Instead of compiling the important formulas (which are available in many other textbooks), we focus on important applications of each distribution in various applied fields like bioinformatics, genomics, ecology, electronics, epidemiology, management, reliability, etc., making this book an indispensable resource for researchers and practitioners in several scientific fields. Examples are drawn from different fields. An up-to-date reference appears at the end of the book. Chapter 1 introduces the basic concepts on random variables, and gives a simple method to find the mean deviation (MD) of discrete distributions. The Bernoulli and binomial distributions are discussed in detail in Chapter 2. A short chapter on discrete uniform distribution appears next. The next two chapters are on geometric and negative binomial distributions. Chapter 6 discusses the Poisson distribution in-depth, including applications in various fields. Chapter 7 is on hypergeometric distribution. As most textbooks in the market either do not discuss, or contain only brief description of the negative hypergeometric distribution, we have included an entire chapter on it. A short chapter on logarithmic series distribution follows it, in which a theorem to find the kth moment of logarithmic distribution using (k-1)th moment of zero-truncated geometric distribution is presented. The last chapter is on multinomial distribution and its applications. The primary users of this book are professionals and practitioners in various fields of engineering and the applied sciences. It will also be of use to graduate students in statistics, research scholars in science disciplines, and teachers of statistics, biostatistics, biotechnology, education, and psychology.

This 1999 book presents single-variable statistical distributions useful in solving practical problems in a wide range of engineering contexts.

This is an introductory book on continuous statistical distributions and its applications. It is primarily written for graduate students in engineering, undergraduate students in statistics, econometrics, and researchers in various fields. The purpose is to give a self-contained introduction to most commonly used classical continuous distributions in two parts. Important applications of each distribution in various applied fields are explored at the end of each chapter. A brief overview of the chapters is as follows. Chapter 1 discusses important concepts on continuous distributions like location-and-scale distributions, truncated, size-biased, and transmuted distributions. A theorem on finding the mean deviation of continuous distributions, and its applications are also discussed. Chapter 2 is on continuous uniform distribution, which is used in generating random numbers from other distributions. Exponential distribution is discussed in Chapter 3, and its applications briefly mentioned. Chapter 4 discusses both Beta-I and Beta-II distributions and their generalizations, as well as applications in geotechnical engineering, PERT, control charts, etc. The arcsine distribution and its variants are discussed in Chapter 5, along with arcsine transforms and Brownian motion. This is followed by gamma distribution and its applications in civil engineering, metallurgy, and reliability. Chapter 7 is on cosine distribution and its applications in signal processing, antenna design, and robotics path planning. Chapter 8 discusses the normal distribution and its variants like lognormal, and skew-normal distributions. The last chapter of Part I is on Cauchy distribution, its variants and applications in thermodynamics, interferometer design, and carbon-nanotube strain sensing. A new volume (Part II) covers inverse Gaussian, Laplace, Pareto, 2, T, F, Weibull, Rayleigh, Maxwell, and Gumbel distributions.

​This is the second part of our book on continuous statistical distributions. It covers inverse-Gaussian, Birnbaum-Saunders, Pareto, Laplace, central 2, , , Weibull, Rayleigh, Maxwell, and extreme value distributions. Important properties of these distribution are documented, and most common practical applications are discussed. This book can be used as a reference material for graduate courses in engineering statistics, mathematical statistics, and econometrics. Professionals and practitioners working in various fields will also find some of the chapters to be useful. Although an extensive literature exists on each of these distributions, we were forced to limit the size of each chapter and the number of references given at the end due to the publishing plan of this book that limits its size. Nevertheless, we gratefully acknowledge the contribution of all those authors whose names have been left out. Some knowledge in introductory algebra and college calculus is assumed throughout the book. Integration is extensively used in several chapters, and many results discussed in Part I (Chapters 1 to 9) of our book are used in this volume. Chapter 10 is on Inverse Gaussian distribution and its extensions. The Birnbaum-Saunders distribution and its extensions along with applications in actuarial sciences is discussed in Chapter 11. Chapter 12 discusses Pareto distribution and its extensions. The Laplace distribution and its applications in navigational errors is discussed in the next chapter. This is followed by central chi-squared distribution and its applications in statistical inference, bioinformatics and genomics. Chapter 15 discusses Student's distribution, its extensions and applications in statistical inference. The distribution and its applications in statistical inference appears next. Chapter 17 is on Weibull distribution and its applications in geology and reliability engineering. Next two chapters are on Rayleigh and Maxwell distributions and its applications in communications, wind energy modeling, kinetic gas theory, nuclear and thermal engineering, and physical chemistry. The last chapter is on Gumbel distribution, its applications in the law of rare exceedances. Suggestions for improvement are welcome. Please send them to [email protected].

A self-contained study of the various applications and developments of discrete distribution theory Written by a well-known researcher in the field, Discrete q-Distributions features an organized presentation of discrete q-distributions based on the stochastic model of a sequence of independent Bernoulli trials. In an effort to keep the book self-contained, the author covers all of the necessary basic q-sequences and q-functions. The book begins with an introduction of the notions of a q-power, a q-factorial, and a q-binomial coefficient and proceeds to discuss the basic q-combinatorics and q-hypergeometric series. Next, the book addresses discrete q-distributions with success probability at a trial varying geometrically, with rate q, either with the number of previous trials or with the number of previous successes. Further, the book examines two interesting stochastic models with success probability at any trial varying geometrically both with the number of trials and the number of successes and presents local and global limit theorems. Discrete q-Distributions also features: Discussions of the definitions and theorems that highlight key concepts and results Several worked examples that illustrate the applications of the presented theory Numerous exercises at varying levels of difficulty that consolidate the concepts and results as well as complement, extend, or generalize the results Detailed hints and answers to all the exercises in an appendix to help less-experienced readers gain a better understanding of the content An up-to-date bibliography that includes the latest trends and advances in the field and provides a collective source for further research An Instructor’s Solutions Manual available on a companion website A unique reference for researchers and practitioners in statistics, mathematics, physics, engineering, and other applied sciences, Discrete q-Distributions is also an appropriate textbook for graduate-level courses in discrete statistical distributions, distribution theory, and combinatorics.

The book provides details on 22 probability distributions. Each distribution section provides a graphical visualization and formulas for distribution parameters, along with distribution formulas. Common statistics such as moments and percentile formulas are followed by likelihood functions and in many cases the derivation of maximum likelihood estimates. Bayesian non-informative and conjugate priors are provided followed by a discussion on the distribution characteristics and applications in reliability engineering.

The long-awaited revision of Fundamentals of Applied Probability and Random Processes expands on the central components that made the first edition a classic. The title is based on the premise that engineers use probability as a modeling tool, and that probability can be applied to the solution of engineering problems. Engineers and students studying probability and random processes also need to analyze data, and thus need some knowledge of statistics. This book is designed to provide students with a thorough grounding in probability and stochastic processes, demonstrate their applicability to real-world problems, and introduce the basics of statistics. The book's clear writing style and homework problems make it ideal for the classroom or for self-study. - Demonstrates concepts with more than 100 illustrations, including 2 dozen new drawings - Expands readers' understanding of disruptive statistics in a new chapter (chapter 8) - Provides new chapter on Introduction to Random Processes with 14 new illustrations and tables explaining key concepts. - Includes two chapters devoted to the two branches of statistics, namely descriptive statistics (chapter 8) and inferential (or inductive) statistics (chapter 9).

This Set Contains: Continuous Multivariate Distributions, Volume 1, Models and Applications, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Continuous Univariate Distributions, Volume 1, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Continuous Univariate Distributions, Volume 2, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Discrete Multivariate Distributions by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Univariate Discrete Distributions, 3rd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Discover the latest advances in discrete distributions theory The Third Edition of the critically acclaimed Univariate Discrete Distributions provides a self-contained, systematic treatment of the theory, derivation, and application of probability distributions for count data. Generalized zeta-function and q-series distributions have been added and are covered in detail. New families of distributions, including Lagrangian-type distributions, are integrated into this thoroughly revised and updated text. Additional applications of univariate discrete distributions are explored to demonstrate the flexibility of this powerful method. A thorough survey of recent statistical literature draws attention to many new distributions and results for the classical distributions. Approximately 450 new references along with several new sections are introduced to reflect the current literature and knowledge of discrete distributions. Beginning with mathematical, probability, and statistical fundamentals, the authors provide clear coverage of the key topics in the field, including: Families of discrete distributions Binomial distribution Poisson distribution Negative binomial distribution Hypergeometric distributions Logarithmic and Lagrangian distributions Mixture distributions Stopped-sum distributions Matching, occupancy, runs, and q-series distributions Parametric regression models and miscellanea Emphasis continues to be placed on the increasing relevance of Bayesian inference to discrete distribution, especially with regard to the binomial and Poisson distributions. New derivations of discrete distributions via stochastic processes and random walks are introduced without unnecessarily complex discussions of stochastic processes. Throughout the Third Edition, extensive information has been added to reflect the new role of computer-based applications. With its thorough coverage and balanced presentation of theory and application, this is an excellent and essential reference for statisticians and mathematicians.

This book introduces descriptive statistics and covers a broad range of topics of interest to students and researchers in various applied science disciplines. This includes measures of location, spread, skewness, and kurtosis; absolute and relative measures; and classification of spread, skewness, and kurtosis measures, L-moment based measures, van Zwet ordering of kurtosis, and multivariate kurtosis. Several novel topics are discussed including the recursive algorithm for sample variance; simplification of complicated summation expressions; updating formulas for sample geometric, harmonic and weighted means; divide-and-conquer algorithms for sample variance and covariance; L-skewness; spectral kurtosis, etc. A large number of exercises are included in each chapter that are drawn from various engineering fields along with examples that are illustrated using the R programming language. Basic concepts are introduced before moving on to computational aspects. Some applications in bioinformatics, finance, metallurgy, pharmacokinetics (PK), solid mechanics, and signal processing are briefly discussed. Every analyst who works with numeric data will find the discussion very illuminating and easy to follow.

The main result of this book is a proof of the contradictory nature of the Navier‒Stokes problem (NSP). It is proved that the NSP is physically wrong, and the solution to the NSP does not exist on R+ (except for the case when the initial velocity and the exterior force are both equal to zero; in this case, the solution (, ) to the NSP exists for all ≥ 0 and (, ) = 0). It is shown that if the initial data 0() ≢ 0, (,) = 0 and the solution to the NSP exists for all ε R+, then 0() := (, 0) = 0. This Paradox proves that the NSP is physically incorrect and mathematically unsolvable, in general. Uniqueness of the solution to the NSP in the space 21(R3) × C(R+) is proved, 21(R3) is the Sobolev space, R+ = [0, ∞). Theory of integral equations and inequalities with hyper-singular kernels is developed. The NSP is reduced to an integral inequality with a hyper-singular kernel.