View source for Template:Acb-w on time

====Confidence Bounds on Time====
<br>
The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:

<br>
::<math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}</math>

<br>
where: 

<br>
::<math>\begin{align}
  {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=&\ \ln (T) \\ 
  z= & \ {{\Phi }^{-1}}\left[ F({T}') \right]  
\end{align}</math>

<br>
and:

<br>
::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

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The next step is to calculate the variance of  <math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}):</math> 

<br>
::<math>\begin{align}
  & Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ 
 &  & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\ 
 &  & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ 
 &  & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)  
\end{align}</math>
 

<br>
or: 

<br>
::<math>\begin{align}
  & Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ 
 &  & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\ 
 &  & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ 
 &  & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)  
\end{align}</math>


<br>
The upper and lower bounds are then found by:


<br>
::<math>\begin{align}
  & T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\ 
 & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')}  
\end{align}</math>

<br>
Solving for  <math>{{T}_{U}}</math>  and  <math>{{T}_{L}}</math>  yields:

<br>
::<math>\begin{align}
  & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ 
 & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}  
\end{align}</math>





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