Exponential Distribution: Its Constructions, Characterizations and Related Distributions

Abstract

Exponential distribution has been constructed in this thesis using transformations of uniform and Pareto distributions. It has also arisen from the Poison process and it is a special case of the gamma distribution. The basic properties of the exponential distribution considered are the r-th moments in general. Derived from the moments are mean, variance, skewness and kurtosis. The moment generating function, cumulant generating function and characteristic function have been stated. There are many ways of characterizing the exponential distribution. In this work we have concentrated on characterization by lack of memory property and its extensions, and, three cases involving order statistics. These are: minimum and spacing between two order statistics, spacing between adjacent order statistics and the k-th order statistic. We have, however, stated other forms of characterizations including many also based on order statistics. Distributions of sum, difference, quotient and product of exponential distributions have been derived. The beta-exponential and the exponentiated exponential distributions have also been derived. These are generalizations of the exponential distribution. Exponential mixtures have been obtained for nine discrete mixing distributions-the Bernoulli, binomial, geometric types I and II, negative binomial types I and II, Poisson, discrete uniform and logarithmic distributions. Mixtures for thirteen continuous mixing distributions have also been determined. These are: beta, exponential, one-parameter gamma, two-parameter gamma, chi-square, inverse gamma, Erlang, inverse Gaussian, generalized inverse-Gaussian, half-normal, Rayleigh, uniform(rectangular) and chi distributions. The survival-time function, hazard rate function, cumulative distribution function and the probability density function have been obtained for each mixture by using the moment generating function technique. Some of the mixture distribution functions were obtained explicitly. Others were obtained in terms of special functions such as modified Bessel, generalized hyper geometric and parabolic cylindrical functions. Density curves with arbitrary parameter values have been sketched for each mixture. An exponential curve, also with arbitrary parameter value, has been superimposed on each mixture density for a rapid visual comparison.