Template:Bounds on time given cumulative mtbf camsaa-cb

Fisher Matrix Bounds
The time, $$T$$, must be positive, thus  $$\ln T$$  is treated as being normally distributed.
 * $$\frac{\ln \hat{T}-\ln T}{\sqrt{Var(\ln \hat{T}})}\ \tilde{\ }\ N(0,1)$$

Confidence bounds on the time are given by:
 * $$CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}$$
 * where:
 * $$\begin{align}

& Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\ & & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda }) \end{align}$$ The variance calculation is the same as Eqn. (variance1) and:
 * $$\hat{T}={{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}$$


 * $$\begin{align}

& \frac{\partial T}{\partial \beta }= & \frac{{{(\lambda \cdot \,{{m}_{c}})}^{1/(1-\beta )}}\ln (\lambda \cdot \text{ }{{m}_{c}})} \\ & \frac{\partial T}{\partial \lambda }= & \frac{\lambda (1-\beta )} \end{align}$$

Crow Bounds

 * Step 1: Calculate $${{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}$$.
 * Step 2: Use the equations from 5.2.8.2 to calculate the bounds on time given the cumulative failure intensity.