A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families

A new way of introducing a parameter to expand a family of distributions is introduced and applied to yield a new two-parameter extension of the exponential distribution which may serve as a competitor to such commonly-used two-parameter families of life distributions as the Weibull, gamma and lognormal distributions. In addition, the general method is applied to yield a new three-parameter Weibull distribution. Families expanded using the method introduced here have the property that the minimum of a geometric number of independent random variables with common distribution in the family has a distribution again in the family. Bivariate versions are also considered.