Template:Weibull Distribution Definition

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 3-parameter Weibull pdf 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]

where [math]\displaystyle{ \beta \,\! }[/math] = shape parameter, [math]\displaystyle{ \eta \,\! }[/math] = scale parameter and [math]\displaystyle{ \gamma\,\! }[/math] = location parameter.

If the location parameter, [math]\displaystyle{ \gamma\,\! }[/math], is assumed to be zero, then the distribution becomes the 2-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 1-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]

For a detailed discussion of this distribution, see The Weibull Distribution.