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| == Weibull Probability Density Function ==
| | #REDIRECT [[The Weibull Distribution]] |
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| === The Three-Parameter Weibull Distribution ===
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| The three-parameter Weibull ''pdf'' is given by:
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| ::<math> f(T)={ \frac{\beta }{\eta }}\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }} </math>
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| :where,
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| ::<math> f(T)\geq 0,\text{ }T\geq 0\text{ or }\gamma, </math>
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| ::<math>\beta>0\ \,\!</math>,
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| ::<math> \eta > 0 \,\!</math>,
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| ::<math> -\infty < \gamma < +\infty \,\!</math>
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| :and,
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| ::<math> \eta= \,\!</math> scale parameter, or characteristic life
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| ::<math> \beta= \,\!</math> shape parameter (or slope),
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| ::<math> \gamma= \,\!</math> location parameter (or failure free life).
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| === The Two-Parameter Weibull Distribution ===
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| The two-parameter Weibull ''pdf'' is obtained by setting
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| <math> \gamma=0 \,\!</math>, and is given by:
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| ::<math> f(T)={ \frac{\beta }{\eta }}\left( {\frac{T}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{T}{\eta }}\right) ^{\beta }} \,\!</math>
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| === The One-Parameter Weibull Distribution ===
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| The one-parameter Weibull ''pdf'' is obtained by again setting
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| <math>\gamma=0 \,\!</math> and assuming <math>\beta=C=Constant \,\!</math> assumed value or:
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| ::<math> f(T)={ \frac{C}{\eta }}\left( {\frac{T}{\eta }}\right) ^{C-1}e^{-\left( {\frac{T}{ \eta }}\right) ^{C}} \,\!</math>
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| where the only unknown parameter is the scale parameter, <math>\eta\,\!</math>.
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| 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.
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