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 <span class="texhtml">β = </span>shape parameter, <span class="texhtml">η = </span>scale parameter and <span class="texhtml">γ</span> =&nbsp;location parameter.
where <math>\beta \,\!</math> = shape parameter, <math>\eta \,\!</math> = scale parameter and <math>\gamma\,\!</math> =&nbsp;location parameter.


If the location parameter, <span class="texhtml">γ</span> , is assumed to be zero, then the distribution becomes the 2-parameter Weibull or:  
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, <span class="texhtml">γ,</span> is zero, and the shape parameter is a known constant, or <span class="texhtml">β = </span>constant <span class="texhtml">= ''C''</span>, so:  
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.