The new class of Kummer beta generalized distributions

Ng and Kotz (1995) introduced a distribution that provides greater flexibility to extremes.We define and study a new class of distributions called the Kummer beta generalized family to extend the normal, Weibull, gamma and Gumbel distributions, among several other well-known distributions. Some special models are discussed. The ordinary moments of any distribution in the new family can be expressed as linear functions of probability weighted moments of the baseline distribution. We examine the asymptotic distributions of the extreme values. We derive the density function of the order statistics, mean absolute deviations and entropies. We use maximum likelihood estimation to fit the distributions in the new class and illustrate its potentiality with an application to a real data set

Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks

Universality is a cornerstone of theories of critical phenomena. It is well understood in most systems especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free energies and response functions is less well understood. The question arises of how universality is maintained in correction-to-scaling in systems of the same universality class but with very different corner geometries. 2D geometries deliver the simplest such examples that can be constructed with and without corners. To investigate how the presence and absence of corners manifest universality, we analyze the spanning tree generating function on two finite systems, namely the cobweb and fan networks. We address how universality can be delivered given that the finite-size cobweb has no corners while the fan has four. To answer, we appeal to the Ivashkevich-Izmailian-Hu approach which unifies the generating functions of distinct networks in terms of a single partition function with twisted boundary conditions. This unified approach shows that the contributions to the individual corner free energies of the fan network sum to zero so that it precisely matches that of the web. Correspondence in each case with results established by alternative means for both networks verifies the soundness of the algorithm. Its range of usefulness is demonstrated by its application to hitherto unsolved problems-namely the exact asymptotic expansions of the logarithms of the generating functions and the conformal partition functions for fan and cobweb geometries. Thus, the resolution of a universality puzzle demonstrates the power of the algorithm and opens up new applications in the future.Comment: This article belongs to the Special Issue Phase Transitions and Emergent Phenomena: How Change Emerges through Basic Probability Models. This special issue is dedicated to the fond memory of Prof. Ian Campbell who has contributed so much to our understanding of phase transitions and emergent phenomen

Primordial Black Holes in non-linear perturbation theory

This thesis begins with a study of the origin of cosmological fluctuations with special attention to those cases in which the non-Gaussian correlation functions are large. The analysis shows that perturbations from an almost massless auxiliary field generically produce large values of the non-linear parameter f_NL. The effects of including non-Gaussian correlation functions in the statistics of cosmological structure are explored by constructing a non-Gaussian probability distribution function (PDF). Such PDF is derived for the comoving curvature perturbation from first principles in the context of quantum field theory, with n-point correlation functions as the only input. The non-Gaussian PDF is then used to explore two important problems in the physics of primordial black holes (PBHs): First, to compute non-Gaussian corrections to the number of PBHs generated from the primordial curvature fluctuations. The second application concerns new cosmological observables. The formation of PBHs is known to depend on two main physical characteristics: the strength of the gravitational field produced by the initial curvature inhomogeneity and the pressure gradient at the edge of the curvature configuration. We account for the probability of finding these configurations by using two parameters: The amplitude of the inhomogeneity and its second radial derivative, evaluated at the centre of the configuration. The implications of the derived probability for the fraction of mass in the universe in the form of PBHs are discussed.Comment: PhD Thesis, Queen Mary, U. of London, Supervisor: Bernard J Carr. (134 pages and 9 figures

Structure of the probability mass function of the Poisson distribution of order kk

The Poisson distribution of order $k$ is a special case of a compound Poisson distribution. For $k=1$ it is the standard Poisson distribution. Although its probability mass function (pmf) is known, what is lacking is a $visual$ interpretation, which a sum over terms with factorial denominators does not supply. Unlike the standard Poisson distribution, the Poisson distribution of order $k$ can display a maximum of $four$ peaks simultaneously, as a function of two parameters: the order $k$ and the rate parameter $\lambda$. This note characterizes the shape of the pmf of the Poisson distribution of order $k$. The pmf can be partitioned into a single point at $n=0$, an increasing sequence for $n \in [1,k]$ and a mountain range for $n>k$ (explained in the text). The ``parameter space'' of the pmf is mapped out and the significance of each domain is explained, in particular the change in behavior of the pmf as a domain boundary is crossed. A simple analogy (admittedly unrelated) is that of the discriminant of a quadratic with real coefficients: its domains characterize the nature of the roots (real or complex), and the domain boundary signifies the presence of a repeated root. Something similar happens with the pmf of the Poisson distribution of order $k$. As an application, this note explains the mode structure of the Poisson distribution of order $k$. Improvements to various inequalities are also derived (sharper bounds, etc.). New conjectured upper and lower bounds for the median and the mode are also proposed.Comment: 33 pages, 10 figure

Обобщение эйлерового интеграла первого рода

Проблематика. У статті запроваджено нове узагальнення ейлерового інтегралу І-го роду (бета-функції), досліджено їх основні властивості. Такі узагальнені функції посідають особливе місце серед спеціальних функцій завдяки їх широкому застосуванню в численних розділах прикладної математики. Мета дослідження. Вивчення нового узагальнення бета-функції та його застосування до обчислення нових інтегралів. Методика реалізації. Для отримання результатів було використано загальні методи теорії спеціальних функцій. Результати дослідження. Запроваджено нове узагальнення ейлеревого інтегралу І-го роду. Для відповідних r-узагальнених бета-функцій було отримано важливі функціональні співвідношення та формули диференціювання. Для широкого застосування в теорії інтегральних і диференціальних рівнянь є суттєвими теореми про зв’язок нових бета-функцій із класичними гіпергеометричними функціями, функціями Макдональда та Віттекера. Висновки. Розглянуте у статті нове узагальнення ейлерового інтегралу І-го роду відкриває широкі можливості для використання ейлерових інтегралів у теорії спеціальних функцій, у прикладних математичних і фізичних задачах. Планується застосувати r-узагальнені бета-функції до розв’язання нових задач теорії ймовірностей, математичної статистики, теорії інтегральних рівнянь тощо.Background. The new generalization of Euler’ integral of the I-kind (beta-functions) is considered, its main properties are investigated. Such distributions have a special place among the special functions due to their widespread use in many areas of applied mathematics. Objective. The aim of the paper is to study the generalization of the new r-generalized beta-function and its application to the calculation of the new integrals. Methods. To obtain results the general methods of the theory of special functions have been used. Results. The article deals with new generalization of Euler’ integral of the I-kind. For the corresponding r-generalized beta functions were obtained important functional relations and differentiation formulas. For a wide application in the theory of integral and differential equations are important theorems on the connection of new beta functions with classical hypergeometric functions, Macdonald’ and Whittaker’ functions. Conclusions. Considered in the article new generalization of Euler’ integral of the I-kind opens up opportunities for the use of Euler’ integrals in the theory of special functions, in the application of mathematical and physical problems. In the future we plan to use r-generalized beta functions to solve the new problems of the theory of probability, mathematical statistics, the theory of integral equations, etc.Проблематика. В статье введено новое обобщение эйлерового интеграла I-го рода (бета-функции), исследованы их основные свойства. Такие обобщенные функции занимают особое место среди специальных функций благодаря их широкому применению в многочисленных разделах прикладной математики. Цель исследования. Изучение нового обобщения бета-функции и его применение к вычислению новых интегралов. Методика реализации. Для получения результатов были использованы общие методы теории специальных функций. Результаты исследования. Введено новое обобщение ейлеревого интеграла I-го рода. Для соответствующих r-обобщенных бета-функций были получены важные функциональные соотношения и формулы дифференцирования. Для широкого применения в теории интегральных и дифференциальных уравнений являются существенными теоремы о связи новых бета-функций с классическими гипергеометрическими функциями, функциями Макдональда и Уиттэкера. Выводы. Рассмотренное в статье новое обобщение эйлерового интеграла I-го рода открывает широкие возможности для использования эйлеровых интегралов в теории специальных функций, в прикладных математических и физических задачах. Планируется применить r-обобщенные бета-функции к решению новых задач теории вероятностей, математической статистики, теории интегральных уравнений и др

Theory of Barnes Beta Distributions

A new family of probability distributions $\beta_{M, N},$ $M=0\cdots N,$ $N\in\mathbb{N}$ on the unit interval $(0, 1]$ is defined by the Mellin transform. The Mellin transform of $\beta_{M, N}$ is characterized in terms of products of ratios of Barnes multiple gamma functions, shown to satisfy a functional equation, and a Shintani-type infinite product factorization. The distribution $\log\beta_{M, N}$ is infinitely divisible. If $M<N,$ $-\log\beta_{M, N}$ is compound Poisson, if $M=N,$ $\log\beta_{M, N}$ is absolutely continuous. The integral moments of $\beta_{M, N}$ are expressed as Selberg-type products of multiple gamma functions. The asymptotic behavior of the Mellin transform is derived and used to prove an inequality involving multiple gamma functions and establish positivity of a class of alternating power series. For application, the Selberg integral is interpreted probabilistically as a transformation of $\beta_{1, 1}$ into a product of $\beta^{-1}_{2, 2}s.$Comment: 15 pages, published version (removed Th. 4.5 and Section 5, updated references

Continuity theorems for the M/M/1/nM/M/1/n queueing system

In this paper continuity theorems are established for the number of losses during a busy period of the $M/M/1/n$ queue. We consider an $M/GI/1/n$ queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution.Comment: Final revision; will be published as i

Generalized Transmuted Family of Distributions: Properties and Applications

We introduce and study general mathematical properties of a new generator of continuous distributions with two extra parameters called the Generalized Transmuted Family of Distributions. We investigate the shapes and present some special models. The new density function can be expressed as a linear combination of exponentiated densities in terms of the same baseline distribution. We obtain explicit expressions for the ordinary and incomplete moments and generating function, Bonferroni and Lorenz curves, asymptotic distribution of the extreme values, Shannon and R´enyi entropies and order statistics, which hold for any baseline model. Further, we introduce a bivariate extension of the new family. We discuss the different methods of estimation of the model parameters and illustrate the potential application of the model via real data. A brief simulation for evaluating Maximum likelihood estimator is done. Finally certain characterziations of our model are presented