Template:Weibull Probability Density Function: Difference between revisions

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== Weibull Probability Density Function ==
#REDIRECT [[The Weibull Distribution]]
 
=== The Three-Parameter Weibull Distribution ===
 
The three-parameter Weibull ''pdf'' is given by:
 
::<math> f(T)={ \frac{\beta }{\eta }}\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }} </math>
 
:where,
 
::<math> f(T)\geq 0,\text{ }T\geq 0\text{ or }\gamma, </math>
 
::<math>\beta>0\ \,\!</math>,
 
::<math> \eta > 0 \,\!</math>,
 
::<math> -\infty < \gamma < +\infty \,\!</math>
 
:and,
 
::<math> \eta= \,\!</math> scale parameter, or characteristic life
::<math> \beta= \,\!</math> shape parameter (or slope),
::<math> \gamma= \,\!</math> location parameter (or failure free life).
 
=== The Two-Parameter Weibull Distribution ===
 
The two-parameter Weibull ''pdf'' is obtained by setting
<math> \gamma=0 \,\!</math>, and is given by:
 
::<math> f(T)={ \frac{\beta }{\eta }}\left( {\frac{T}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{T}{\eta }}\right) ^{\beta }} \,\!</math>
 
=== The One-Parameter Weibull Distribution ===
 
The one-parameter Weibull ''pdf'' is obtained by again setting
<math>\gamma=0 \,\!</math> and assuming <math>\beta=C=Constant \,\!</math> assumed value or:
 
::<math> f(T)={ \frac{C}{\eta }}\left( {\frac{T}{\eta }}\right) ^{C-1}e^{-\left( {\frac{T}{ \eta }}\right) ^{C}} \,\!</math>
 
where the only unknown parameter is the scale parameter, <math>\eta\,\!</math>.
 
Note that in the formulation of the one-parameter Weibull, we assume that the shape parameter <math>\beta \,\!</math> is known ''a priori'' from past experience on identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Latest revision as of 01:27, 13 August 2012

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