Template:Bounds on Time and Reliability.LRCB.FMB.ED

Bounds on Time and Reliability
In order to calculate the bounds on a time estimate for a given reliability, or on a reliability estimate for a given time, the likelihood function needs to be rewritten in terms of one parameter and time/reliability, so that the maximum and minimum values of the time can be observed as the parameter is varied. This can be accomplished by substituting a form of the exponential reliability equation into the likelihood function. The exponential reliability equation can be written as:


 * $$R={{e}^{-\lambda \cdot t}}$$

This can be rearranged to the form:


 * $$\lambda =\frac{-\text{ln}(R)}{t}$$

This equation can now be substituted into the likelihood ratio equation to produce a likelihood equation in terms of $$t$$ and $$R:$$


 * $$L(t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\left( \frac{-\text{ln}(R)}{t} \right)\cdot {{e}^{\left( \tfrac{\text{ln}(R)}{t} \right)\cdot {{x}_{i}}}}$$

The unknown parameter $$t/R$$ depends on what type of bounds are being determined. If one is trying to determine the bounds on time for the equation for the mean and the Bayes' rule equation for single parametera given reliability, then $$R$$ is a known constant and $$t$$ is the unknown parameter. Conversely, if one is trying to determine the bounds on reliability for a given time, then $$t$$ is a known constant and $$R$$ is the unknown parameter. Either way, the likelihood ratio function can be solved for the values of interest.

Example 6:

Example 7: