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:

[math]\displaystyle{ R={{e}^{-\lambda \cdot t}} }[/math]

This can be rearranged to the form:

[math]\displaystyle{ \lambda =\frac{-\text{ln}(R)}{t} }[/math]

This equation can now be substituted into Eqn. (explikelihood) to produce a likelihood equation in terms of [math]\displaystyle{ t }[/math] and [math]\displaystyle{ R\ \ : }[/math]

[math]\displaystyle{ 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}}}} }[/math]

The unknown parameter [math]\displaystyle{ t/R }[/math] depends on what type of bounds are being determined. If one is trying to determine the bounds on time for a given reliability, then [math]\displaystyle{ R }[/math] is a known constant and [math]\displaystyle{ t }[/math] is the unknown parameter. Conversely, if one is trying to determine the bounds on reliability for a given time, then [math]\displaystyle{ t }[/math] is a known constant and [math]\displaystyle{ R }[/math] is the unknown parameter. Either way, Eqn. (expliketr) can be used to solve Eqn. (lratio3) for the values of interest.