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