Template:Weibull Distribution Definition: Difference between revisions
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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> | ::<math>f(t)=\frac{\beta}{\eta } \left( \frac{t-\gamma }{\eta } \right)^{\beta -1}{e}^{-(\tfrac{t-\gamma }{\eta }) ^{\beta}}</math> | ||
where < | where <math>\beta \,\!</math> = shape parameter, <math>\eta \,\!</math> = scale parameter and <math>\gamma\,\!</math> = location parameter. | ||
If the location parameter, < | If the location parameter, <math>\gamma\,\!</math> , is assumed to be zero, then the distribution becomes the 2-parameter Weibull or: | ||
::<math>f(t)=\frac{\beta}{\eta }( \frac{t }{\eta } )^{\beta -1}{e}^{-(\tfrac{t }{\eta }) ^{\beta}}</math> | ::<math>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, < | One additional form is the 1-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: | ||
::<math>f(t)=\frac{C}{\eta}(\frac{t}{\eta})^{C-1}e^{-(\frac{t}{\eta})^C} | ::<math>f(t)=\frac{C}{\eta}(\frac{t}{\eta})^{C-1}e^{-(\frac{t}{\eta})^C} | ||
Revision as of 00:34, 22 August 2012
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 [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]
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.