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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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| <br>
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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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| <br>
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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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| <br>
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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:
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| ::<math>f(t)=\frac{\beta}{\eta }( \frac{t }{\eta } )^{\beta -1}{e}^{-(\tfrac{t }{\eta }) ^{\beta}}</math>
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| 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:
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| ::<math>f(t)=\frac{C}{\eta}(\frac{t}{\eta})^{C-1}e^{-(\frac{t}{\eta})^C}
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| </math>
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| Chapter 6 of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.
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| <br>
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| ====The Weibull-Bayesian Distribution====
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| 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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| <br>
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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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| <br>
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| The Weibull-Bayesian distribution and its characteristics are presented in more detail in Chapter 6.
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