Failure Rate and Mean Remaining Lifetime

Abstract

Reliability engineering, survival analysis and other disciplines mostly deal with positive random variables, which are often called lifetimes. As a random variable, a lifetime is completely characterized by its distribution function. A realization of a lifetime is usually manifested by a failure, death or some other ‘end event’. Therefore, for example, information on the probability of failure of an operating item in the next (usually sufficiently small) interval of time is really important in reliability analysis. The failure (hazard) rate function λ(t) defines this probability of interest. If this function is increasing, then our object is usually degrading in some suitable probabilistic sense, as the conditional probability of failure in the corresponding infinitesimal interval of time increases with time. For example, it is well known that the failure (mortality) rate of adult humans increases exponentially with time; the failure rate of many mechanically wearing devices is also increasing. Thus, understanding and analysing the shape of the failure rate is an essential part of reliability and lifetime data analysis. Similar to the distribution function F(t) , the failure rate also completely characterizes the corresponding random variable. It is well known that there exists a simple, meaningful exponential representation for the absolutely continuous distribution function in terms of the corresponding failure rate (Section 2.1).KeywordsFailure RateSurvival FunctionLifetime DistributionInverse Gaussian DistributionErlangian DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.