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| ===Eyring-Weibull Statistical Properties Summary===
| | #REDIRECT [[Eyring_Relationship#Eyring-Weibull]] |
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| ====Mean or MTTF====
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| The mean, <math>\overline{T}</math>, or Mean Time To Failure (MTTF) for the Eyring-Weibull model is given by:
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| ::<math>\overline{T}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)</math>
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| where <math>\Gamma \left( \tfrac{1}{\beta }+1 \right)</math> is the gamma function evaluated at the value of <math>\left( \tfrac{1}{\beta }+1 \right)</math> .
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| <br>
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| ====Median====
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| <br>
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| The median, <math>\breve{T}</math>
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| for the Eyring-Weibull model is given by:
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| <br>
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| ::<math>\breve{T}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
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| ====Mode====
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| <br>
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| The mode, <math>\tilde{T},</math>
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| for the Eyring-Weibull model is given by:
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| <br>
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| ::<math>\tilde{T}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>
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| <br>
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| ====Standard Deviation====
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| <br>
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| The standard deviation, <math>{{\sigma }_{T}},</math>
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| for the Eyring-Weibull model is given by:
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| <br>
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| ::<math>{{\sigma }_{T}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}</math>
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| <br>
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| ====Eyring-Weibull Reliability Function====
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| <br>
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| The Eyring-Weibull reliability function is given by:
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| <br>
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| ::<math>R(T,V)={{e}^{-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}</math>
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| <br>
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| ====Conditional Reliability Function====
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| <br>
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| The Eyring-Weibull conditional reliability function at a specified stress level is given by:
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| <br>
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| ::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \left( T+t \right)\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}}{{{e}^{-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>R(T,t,V)={{e}^{-\left[ {{\left( \left( T+t \right)\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }} \right]}}</math>
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| <br>
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| ====Reliable Life====
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| <br>
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| For the Eyring-Weibull model, the reliable life, <math>{{t}_{R}}</math> , of a unit for a specified reliability and starting the mission at age zero is given by:
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| <br>
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| ::<math>{{t}_{R}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}{{\left\{ -\ln \left[ R\left( {{T}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>
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| ====Eyring-Weibull Failure Rate Function====
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| <br>
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| The Eyring-Weibull failure rate function, <math>\lambda (T)</math> , is given by:
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| <br>
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| ::<math>\lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\beta {{\left( T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta -1}}</math>
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| <br>
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