A STUDY OF SOME TYPES OF BETA DISTRIBUTION

Abstract

The beta distribution is one of the distributions that are applicable in different aspects, such as rule of succession and task duration modeling. Examples of events that may be modeled by beta distribution include: the time taken to complete a task and the proportion of defective items in a shipment. The beta distributions are used extensively in Bayesian statistics, since beta distributions provide a family of conjugate prior distributions for binomial and geometric distributions. Bivariate beta distributions are used in a wide variety of applications such as Bayesian statistics and reliability theory. They form part of the Dirichlet family of distributions, but have become an important family by itself. Many bivariate beta distributions have been derived out of applications, as extensions, or generalizations from other well-known bivariate beta distributions. This thesis aims to study the Connor and Mosimann's bivariate Dirichlet distribution, where the variables of the distribution, say and may be considered as proportions. This distribution has several applications in many areas, such as statistical decision theory, contingency tables, geochemistry and forensic sciences. Data comprising of proportions arise in most areas of sciences and engineering. Some examples of data set that involve proportions are: sand, silt, clay compositions of sediment samples and compositions of lavas. Linear combinations, ratios and products of two random variables have attracted many researchers in the statistics literature due to its practical importance in the fields of reliability engineering, industrial engineering and computer systems. The random variables may be independent or dependent. In this thesis, we derive the exact distributions of the difference , ratio and product of two dependent random variables and following the Connor and Mosimann's bivariate Dirichlet distribution. Some statistical characteristics of these distributions are studied, such as the moments and the hazard rate function. As the exact distributions of and derived have complex forms involving the Gauss hypergeometric and the Appell function ,we deduce approximate distributions for these variables depending on beta type I and type II distributions.