Template:WeibullDistribution

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]\displaystyle{ pdf }[/math] is defined by:

[math]\displaystyle{ f(t)=\frac{\beta}{\eta } \left( \frac{t-\gamma }{\eta } \right)^{\beta -1}{e}^{-(\tfrac{t-\gamma }{\eta }) ^{\beta}} }[/math]

with three parameters [math]\displaystyle{ \beta }[/math] , [math]\displaystyle{ \eta }[/math] and [math]\displaystyle{ \gamma , }[/math] where [math]\displaystyle{ \beta = }[/math] shape parameter, [math]\displaystyle{ \eta = }[/math] scale parameter and location parameter.
If the location parameter, [math]\displaystyle{ \gamma }[/math] , is assumed to be zero, the distribution then becomes the two-parameter Weibull or:

[math]\displaystyle{ 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]\displaystyle{ \gamma , }[/math] is zero, and the shape parameter is a known constant, or [math]\displaystyle{ \beta = }[/math] constant [math]\displaystyle{ =C }[/math], so:

[math]\displaystyle{ 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++.

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]\displaystyle{ \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.
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.
The Weibull-Bayesian distribution and its characteristics are presented in more detail in Chapter 6.