Adding a parameter to the exponential and Weibull distributions with applications

A generalization of the exponential distribution is studied. This new distribution is the natural conjugate prior for the continuous Lindley distribution. Since this distribution belongs to the natural exponential family of distributions, it has sufficient fixed-dimension statistics for varying sample sizes, and a conjugate prior distribution exists. The result obtained is a generalization of the exponential distribution which is applied in credibility theory and in other settings. The properties of this distribution and a generalization of the two-parameter Weibull distribution obtained from it are also presented.