The beta family of distributions, whose origin can be traced to 1676 in a letter from Sir Issac Newton to Henry Oldenbeg [Dutka, J., 1981, The incomplete beta function–A historic profile. Archive for History of Exact Sciences, 24, 11–29.], has been utilized extensively in statistical theory and practice for over a century. Originally defined on the unit interval [0, 1] but easily extended to any finite interval, the beta distributions can take an amazingly great variety of forms. They can be fitted practically to any data representing a phenomenon in almost any field of application. In this paper, we provide a review of the properties and the variations of beta distributions as well as their relationship to other distributions. We also illustrate two real life applications to economic and drought data sets.

Multitude of beta distributions with applications

This paper presents a compound of the generalized negative binomial distribution with the generalized beta distribution.

A compound of the generalized negative binomial distribution with the generalized beta distribution

To any (0,2)-tensor field on the tangent and cotangent bundles of a semi-Riemannian manifold, we associate a global matrix function ‘mutatis mutandis’ as in the Riemannian case. Based on this fact, natural (0,2)tensor fields on these bundles are defined and characterized.

Natural tensor fields of type $(0,2)$ on the tangent and cotangent bundles of a semi-Riemannian manifold

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Limit Property of a Compound of the Generalized Negative Binomial and Beta Distributions

A Compound of an Inflated Pascal Distribution with the Poisson One

In paper [6] J.G. Skellam gave a distribution which grew out of a compound of a binomial distribution and a beta one. In the present paper we consider an analogous problem by compounding a binomial distribution and a generalized beta one given in [8] by T. Środka. The distribution obtained yields interesting special as well as limit cases constituting compounds of the binomial distribution and those most frequently encountered in statistical practice.

Tadeusz Gerstenkorn

Papers

The Compounding of the Binomial and Generalized Beta Distributions