Properties of distributions with increasing failure rate

This paper solves the search for interior solutions to optimization problems using stochastic variables. This is done by way of some new properties of distribution functions with increasing failure rates as characterized in Barlow and Proschan (1965). Building upon Lariviere (2006), we show that an objective function of the type R(x) = F(x)+xF(x), where F(x) = 1-F(x), can also admit one interior maximal solution when the distribution function F has an increasing failure rate (IFR).