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| ===Fisher Matrix Bounds===
| | #REDIRECT [[The Exponential Distribution]] |
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| {{Bounds on the Parameters FMB ED}}
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| ====Bounds on Reliability====
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| The reliability of the two-parameter exponential distribution is: | |
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| ::<math>\hat{R}(T;\hat{\lambda })={{e}^{-\hat{\lambda }(T-\hat{\gamma })}}</math>
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| The corresponding confidence bounds are estimated from:
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| ::<math>\begin{align}
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| & {{R}_{L}}= & {{e}^{-{{\lambda }_{U}}(T-\hat{\gamma })}} \\
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| & {{R}_{U}}= & {{e}^{-{{\lambda }_{L}}(T-\hat{\gamma })}}
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| \end{align}</math>
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| These equations hold true for the one-parameter exponential distribution, with <math>\gamma =0</math>.
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| ====Bounds on Time====
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| The bounds around time for a given exponential percentile, or reliability value, are estimated by first solving the reliability equation with respect to time, or reliable life:
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| ::<math>\hat{T}=-\frac{1}{{\hat{\lambda }}}\cdot \ln (R)+\hat{\gamma }</math>
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| The corresponding confidence bounds are estimated from:
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| ::<math>\begin{align}
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| & {{T}_{U}}= & -\frac{1}{{{\lambda }_{L}}}\cdot \ln (R)+\hat{\gamma } \\
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| & {{T}_{L}}= & -\frac{1}{{{\lambda }_{U}}}\cdot \ln (R)+\hat{\gamma }
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| \end{align}</math>
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| The same equations apply for the one-parameter exponential with <math>\gamma =0.</math>
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