## Theses, Dissertations and Capstones

#### Title

The Quotient of the beta-Weibull Distribution

2012

Mathematics

#### College

College of Science

M.A.

Thesis

Alfred Akinsete

Ari Aluthge

#### Abstract

A new class of distributions recently developed involves the logit of the beta distribution. Among this class of distributions are, the beta-Normal (Eugene et al. ); beta-Gumbel (Nadarajah and Kotz ); beta-Exponential (Nadarajah and Kotz ); beta-Weibull (Famoye et al. ); beta-Rayleigh (Akinsete and Lowe ); beta-Laplace (Kozubowshi and Nadarajah ); and beta-Pareto (Akinsete et al. ), among a few others. Many useful statistical properties arising from these distributions and their applications to real life data have been discussed in literature. One approach by which a new statistical distribution is generated is by the transformation of random variables having known distribution function(s). The focus of this work is to investigate the statistical properties of the quotient of the beta-Weibull distribution. The latter was defined and extensively studied by Famoye et al. . That is, if X and Y are random variables having a beta-Weibull distribution with parameters a1, B1, c1, and Y1 and a2, B2, c2, and Y2 respectively, i.e. X-BW (a1, B1, c1, and Y1) and Y-BW( a2, B2, c2, and Y2), what then is the distribution of the quotient of X and Y? That is, the distribution of the random variable V = X/Y. We obtain the probability density function (pdf) and the cumulative distribution (cdf) of this distribution. Various statistics of the distribution are obtained, including, for example, moments, moment and characteristic generating functions, hazard function, and the entropy. We propose the method of Maximum Likelihood Estimator (MLE) for estimating the parameters of the distribution. The open source software R and Python are used extensively in implementing our results.

#### Subject(s)

Weibull distribution.

Statistics.

COinS