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.
Read more about this topic: Failure Rate
Famous quotes containing the words failure, rate, continuous and/or sense:
“In the deeper layers of the modern consciousness ... every attempt to succeed is an act of aggression, leaving one alone and guilty and defenseless among enemies: one is punished for success. This is our intolerable dilemma: that failure is a kind of death and success is evil and dangerous, is—ultimately—impossible.”
—Robert Warshow (1917–1955)
“Strange that the vanity which accompanies beauty—excusable, perhaps, when there is such great beauty, or at any rate understandable—should persist after the beauty was gone.”
—Mary A. [Elizabeth, Countess Von] Arnim (1866–1941)
“There is no such thing as a life of passion any more than a continuous earthquake, or an eternal fever. Besides, who would ever shave themselves in such a state?”
—George Gordon Noel Byron (1788–1824)
“Books, the oldest and the best, stand naturally and rightfully on the shelves of every cottage. They have no cause of their own to plead, but while they enlighten and sustain the reader his common sense will not refuse them. Their authors are a natural and irresistible aristocracy in every society, and, more than kings or emperors, exert an influence on mankind.”
—Henry David Thoreau (1817–1862)