On the mean past and the mean residual life under double monitoring

In the present article, we consider a parallel system consisting of n identical components, such that the lifetimes of components are independent and have a common distribution function F. It is assumed that the total number of failures of the components at time t(1) is m and at time t(2) (t(1)<t(2)), all components of the system have failed or the system is still working. Under these conditions, we are interested in the study of the mean past lifetime (MPL) of the components and the mean residual lifetime (MRL) of the system. Several properties of the MPL and MRL are studied and several properties of those are derived. It is also shown that the underlying distribution function F can be recovered from the proposed MPL and a characterization of the exponential distribution is given based on MRL.