J. F. Lawless [6] and G. S. Lingappaiah [8] constructed prediction intervals for future failure times based on times of early failures when life is exponentially distributed. Some aspects of these interesting papers are discussed here. Of special interest is the amount of information for prediction contained in the latest available failure time, relative to the amount of information in all of the failure times up to the latest one. It is found that this relative information is very high in small samples as well as asymptotically, regardless of the time of prediction or the time to be predicted. Recurrence relations for certain distribution functions which arise in connection with the prediction intervals are given and large sample properties of the prediction intervals of Lawless and Lingappaiah are derived. Conditions for the best linear unbiased predictor to be the best unbiased predictor are discussed.