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 $$pdf$$ is defined by:
 * $$f(t)=\frac{\beta}{\eta } \left( \frac{t-\gamma }{\eta } \right)^{\beta -1}{e}^{-(\tfrac{t-\gamma }{\eta }) ^{\beta}}$$

with three parameters $$\beta $$,  $$\eta $$  and  $$\gamma ,$$  where  $$\beta =$$  shape parameter,  $$\eta =$$  scale parameter and location parameter. If the location parameter, $$\gamma $$, is assumed to be zero, the distribution then becomes the two-parameter Weibull or:


 * $$f(t)=\frac{\beta}{\eta }( \frac{t }{\eta } )^{\beta -1}{e}^{-(\tfrac{t }{\eta }) ^{\beta}}$$

One additional form is the one-parameter Weibull distribution, which assumes that the location parameter, $$\gamma ,$$ is zero, and the shape parameter is a known constant, or $$\beta =$$ constant $$=C$$, so:


 * $$f(t)=\frac{C}{\eta}(\frac{t}{\eta})^{C-1}e^{-(\frac{t}{\eta})^C}

$$

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 ( $$\beta )$$ 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.