Template:Eyring-weibull stat prop sum: Difference between revisions

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(Created page with '===Eyring-Weibull Statistical Properties Summary=== ====Mean or MTTF==== The mean, <math>\overline{T}</math>, or Mean Time To Failure (MTTF) for the Eyring-Weibull model is gi…')
 
 
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===Eyring-Weibull Statistical Properties Summary===
#REDIRECT [[Eyring_Relationship#Eyring-Weibull]]
 
====Mean or MTTF====
 
The mean,  <math>\overline{T}</math>, or Mean Time To Failure (MTTF) for the Eyring-Weibull model is given by:
 
 
::<math>\overline{T}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)</math>
 
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> .
 
<br>
 
====Median====
 
<br>
The median,  <math>\breve{T}</math>
for the Eyring-Weibull model is given by:
 
<br>
::<math>\breve{T}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
 
====Mode====
 
<br>
The mode,  <math>\tilde{T},</math>
for the Eyring-Weibull model is given by:
 
<br>
::<math>\tilde{T}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>
 
<br>
 
====Standard Deviation====
 
<br>
The standard deviation,  <math>{{\sigma }_{T}},</math> 
for the Eyring-Weibull model is given by:
 
<br>
::<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>
 
<br>
====Eyring-Weibull Reliability Function====
 
<br>
The Eyring-Weibull reliability function is given by:
 
<br>
::<math>R(T,V)={{e}^{-{{\left( V\cdot T\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}} \right)}^{\beta }}}}</math>
<br>
====Conditional Reliability Function====
<br>
 
The Eyring-Weibull conditional reliability function at a specified stress level is given by:
<br>
::<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>
<br>
:or:
 
<br>
::<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>
<br>
====Reliable Life====
<br>
 
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:
 
<br>
::<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>
 
 
====Eyring-Weibull Failure Rate Function====
<br>
The Eyring-Weibull failure rate function,  <math>\lambda (T)</math> , is given by:
<br>
 
::<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>
<br>

Latest revision as of 23:14, 16 August 2012

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