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Life Distributions (Page 4 of 4)
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The Weibull Life Distribution
The Weibull life distribution was developed by W. Weibull of Sweden to investigate metal fatigue failures. It is described by a location parameter a and a shape factor b, and is similar to the lognormal distribution in many ways. Two of the major differences between them are: 1) the Weibull distribution's probability density function does not start from zero; and 2) its failure rate curve l(t) is monotonically increasing for b > 1 and monotonically decreasing for b < 1.
The Weibull distribution can take on many shapes, depending on the value of the shape factor b. In fact, by varying the value of b, all the phases of the bath tub curve can be modeled by the Weibull distribution. The early life phase, wherein the failure rate decreases with time, can be represented by the Weibull distribution with b < 1. The steady-state phase, wherein the failure rate is constant, can be represented by the Weibull distribution with b = 1. Finally, letting b be > 1 will make the Weibull distribution a model for the wear-out phase, wherein the failure rate increases with time.
Figure 4. The f(t), F(t), and l(t) of a Weibull life distribution; source: D. S. Peck and O. D. Trapp, Accelerated Testing Handbook, Technology Associates.
The Weibull distribution has become popular in reliability engineering, partly because of its simpler math and flexibility, and partly because earlier works using this distribution have found it to fit some failure mechanisms nicely. A closer look at the same mechanisms showed that they, too, fit the lognormal distribution.
Thus, the lognormal distribution should have been a better choice in the first place since its mathematics are consistent with the physical phenomena taking place. Care must therefore be taken when an engineer sees data fitting the Weibull distribution, since they can turn out to be lognormal in reality.
Please see Life Distribution Functions for more detailed mathematical descriptions of these life distributions.
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