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| === The Mean or MTTF ===
| | #REDIRECT [[Weibull Distribution Functions]] |
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| The mean, <math> \overline{T} \,\!</math>, (also called ''MTTF'' of the Weibull ''pdf'' is given by:
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| ::<math> \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\!</math>
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| where
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| ::<math> \Gamma \left( {\frac{1}{\beta }}+1\right) \,\!</math>
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| is the gamma function evaluated at the value of
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| ::<math> \left( { \frac{1}{\beta }}+1\right) \,\!</math>.
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| The gamma function is defined as:
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| ::<math> \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\!</math>
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| For the 2-parameter case, this can be reduced to:
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| ::<math> \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\!</math>
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| Note that some practitioners erroneously assume that <math> \eta \,\!</math> is equal to the MTTF, <math> \overline{T}\,\!</math>.
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| This is only true for the case of
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| <math> \beta=1 \,\!</math> or
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| ::<math>
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| \begin{align}
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| \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\
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| &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\
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| &= \eta \cdot \Gamma \left( {2}\right) \\
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| &= \eta \cdot 1\\
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| &= \eta
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| \end{align}
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| </math>
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