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 8 of this reference fully details the Weibull distribution and presents many examples of its use in Weibull++.