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 Eqn. (explikelihood) 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 a 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.