Qual Quant (2009) 43:855–863
Bathtub-shaped failure rate functions
Published online: 4 January 2008
© Springer Science+Business Media B.V. 2007
Abstract The failure rate function is an important quantity characterizing life phenomena.
Ideally, one would expect this function to exhibit a bathtub shape. In this paper, a compre-
hensive review of the known distributions that exhibit this shape is provided. Over 17 such
distributions are identiﬁed. This review is especially important because almost all of the
commonly known distributions in statistics do not exhibit the bathtub shape. Furthermore, it
could serve as an important reference and encourage developments of further distributions
that exhibit a bathtub shape.
Keywords Bathtub shape · Failure rate function · Weibull distribution
Let X be a random variable representing the lifetime of some system with the probability
density function (pdf) f (x ) and the cumulative distribution function (cdf) F (x ). The failure
rate function of X is deﬁned by
h(x ) =
f (x )
1 − F(x )
The failure rate function is an important quantity characterizing life phenomena. It can be
loosely interpreted as the conditional probability of failure, given survival to the time x.
Ideally one would like a “bathtub” shape for h(x ) that captures the three distinct hazard
regimes: the region of infant mortality (where h(x ) decreases with x), the random failure
region (where h(x ) does not change rapidly with x) and the wear-out region (where h(x )
increases with x due to deterioration processes).
Almost all of the standard distributions in statistics do not exhibit a bathtub shape for h(x ).
Even the traditional Weibull distribution does not exhibit a bathtub shape for h(x ).Thus,it
is important that one knows which distributions exhibit this shape. This will be important
S. Nadarajah (
School of Mathematics, University of Manchester, Manchester M60 1QD, UK