``Setting the clock back to zero'' property of a family of life distributions (Q918617)

We define, what we have called the ``setting the clock back to zero'' or an invariance property of a family of life distributions. This property ensures that the conditional distribution of the additional time of survival of a living organism or an industrial item, given that it has survived \(x_ 0\) units, remains in the family. This concept generalizes the lack of memory property of the exponential distribution, for which the conditional distribution of additional survival time is exactly the same as the original distribution.