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===The Weibull Distribution===
#REDIRECT [[The Weibull Distribution]]
The Weibull distribution is a general purpose reliability distribution used to model material strength, times-to-failure of electronic and mechanical components, equipment or systems. In its most general case, the three-parameter Weibull <math>pdf</math> is defined by:
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::<math>f(t)=\frac{\beta}{\eta } \left( \frac{t-\gamma }{\eta } \right)^{\beta -1}{e}^{-(\tfrac{t-\gamma }{\eta }) ^{\beta}}</math>
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with three parameters  <math>\beta </math> ,  <math>\eta </math>  and  <math>\gamma ,</math>  where  <math>\beta =</math>  shape parameter,  <math>\eta =</math>  scale parameter and location parameter.
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If the location parameter,  <math>\gamma </math> , is assumed to be zero, the distribution then becomes the two-parameter Weibull or:
 
::<math>f(t)=\frac{\beta}{\eta }( \frac{t }{\eta } )^{\beta -1}{e}^{-(\tfrac{t }{\eta }) ^{\beta}}</math>
 
One additional form is the one-parameter Weibull distribution, which assumes that the location parameter, <math>\gamma ,</math> is zero, and the shape parameter is a known constant, or <math>\beta =</math> constant <math>=C</math>, so:
 
::<math>f(t)=\frac{C}{\eta}(\frac{t}{\eta})^{C-1}e^{-(\frac{t}{\eta})^C}
</math>
 
Chapter 6 of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.
 
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====The Weibull-Bayesian Distribution====
Another approach is the Weibull-Bayesian model which assumes that the analyst has some prior knowledge about the distribution of the shape parameter ( <math>\beta )</math>  of the Weibull distribution. There are many practical applications for this model, particularly when dealing with small sample sizes and/or some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data or physics-of-failure.
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Note that this is not the same as the so called WeiBayes model.  The so called WeiBayes model is really a one-parameter Weibull distribution. It assumes a fixed value (constant) for the shape parameter and solves for the scale parameter.  The Weibull-Bayesian model in Weibull++ 7 is actually a true WeiBayes model and offers an alternative to the one-parameter Weibull by including the variation and uncertainty that is present in the prior estimation of the shape parameter.
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The Weibull-Bayesian distribution and its characteristics are presented in more detail in Chapter 6.

Latest revision as of 10:12, 9 August 2012

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