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| ====Bounds on Time and Reliability====
| | #REDIRECT [[The Exponential Distribution]] |
| 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:
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| ::<math>R={{e}^{-\lambda \cdot t}}</math>
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| This can be rearranged to the form:
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| ::<math>\lambda =\frac{-\text{ln}(R)}{t}</math>
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| This equation can now be substituted into Eqn. (explikelihood) to produce a likelihood equation in terms of <math>t</math> and <math>R\ \ :</math>
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| ::<math>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>
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| The unknown parameter <math>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>R</math> is a known constant and <math>t</math> is the unknown parameter. Conversely, if one is trying to determine the bounds on reliability for a given time, then <math>t</math> is a known constant and <math>R</math> is the unknown parameter. Either way, Eqn. (expliketr) can be used to solve Eqn. (lratio3) for the values of interest.
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| {{Exponential Distribution: Likelihood Ratio Bound for Time}}
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| {{Exponential Distribution: Likelihood Ratio Bound for Reliability}}
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