Failure Rate in The Continuous Sense
Calculating the failure rate for ever smaller intervals of time, results in the hazard function (loosely and incorrectly called hazard rate), . This becomes the instantaneous failure rate as tends to zero:
A continuous failure rate depends on the existence of a failure distribution, which is a cumulative distribution function that describes the probability of failure (at least) up to and including time t,
where is the failure time. The failure distribution function is the integral of the failure density function, f(t),
The hazard function can be defined now as
Many probability distributions can be used to model the failure distribution (see List of important probability distributions). A common model is the exponential failure distribution,
which is based on the exponential density function. The hazard rate function for this is:
Thus, for an exponential failure distribution, the hazard rate is a constant with respect to time (that is, the distribution is "memory-less"). For other distributions, such as a Weibull distribution or a log-normal distribution, the hazard function may not be constant with respect to time. For some such as the deterministic distribution it is monotonic increasing (analogous to "wearing out"), for others such as the Pareto distribution it is monotonic decreasing (analogous to "burning in"), while for many it is not monotonic.
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