Mixture of distributions as a lifetime distribution of a bus engine

We show that a upside-down bathtub failure rate function can be obtained from a mixture of two increasing failure rate function (IFR) models. Specifically, we study the failure rate of the mixture an exponential distribution, and an IFR distribution with strictly increasing failure rate function. Examples of several other upside-down bathtub shaped failure rate functions are also presented. The method are illustrated by a numerical example of the time between the failures for the bus engines.