Two problems involving the derivation of bounds on distributions with a decreasing failure rate (DFR distributions) are presented. Given that an item has a decreasing failure rate, sharp upper and lower bounds on the burn-in time to achieve a specified mean residual life are derived The bounds rely only on the DFR assumption and a knowledge of the first moment and a percentile of the failure distribution. An early estimate of the five year survival proportion (com.monly called the five year

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... rate) is of great interest in assessing the value of a treatment for a mortal disease such as cancer. Assuming that the distribution 01 time to death is DFR and assuming a knowledge of the mean and a percentile, sharp upper and lower bounds on the survival proportion are obtained. In addition some bounds on the hazard rate and density of a DFR distribution are given.