Bounds are obtained for conditional moments of random variablesX, such asE(X|X>a), and of order statistics from members of monotone failure rate families of distributions. Numerical bounds for Weibull and other families are calculated and compared. Results classifying some functions of random variables as having monotone failure rate distributions are obtained, as are characterizations of absolutely continuous distributions with cdf's of the form 1 – exp[–H(x)],x> 0.