Bathtub shaped failure rates from mixtures: a practical point of view

We show that a bathtub shaped failure rate can be obtained from a mixture of two increasing failure rate (IFR) models. Specifically, we study the failure rate of the mixture of an exponential distribution, and a Weibull distribution with strictly increasing failure rate. Under some reasonable conditions, we show that, from a practical point of view, the mixture failure rate is bathtub. Similar results can be obtained from other mixtures.