Case 1: (Generalized Exponential Distribution (GED)) non-Bayesian case

Generalized Exponential (GE) distribution is a three parameter distribution in general. Out of the three parameters, the two parameters for GE distribution are the same as in Gamma and Weibull distribution and they are the scale () and the shape () parameters. Gamma distribution has been extensively used for analyzing the lifetime data both for its increasing as well as decreasing failure rates which depends upon the shape parameter (), but the non existence of any closed form expression for its distribution function resulted in Weibull distribution being used more frequently, since the latter has a closed form expression for its corresponding distribution function. Furthermore GE distribution was given as an option for the Gamma and Weibull distributions and the interesting fact is that this distribution function has closed form expression. GE distribution has likelihood ratio ordering with respect to the shape parameter when the scale parameter is constant, this property is same as that for the Gamma distribution.

The the three parameter GE along with its survival and hazard function formulae are given as follows