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| | #REDIRECT [[The Exponential Distribution]] |
| ===Bayesian Confidence Bounds===
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| ====Bounds on Parameters====
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| From Chapter [[Confidence Bounds]], we know that the posterior distribution of <math>\lambda </math> can be written as:
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| ::<math>f(\lambda |Data)=\frac{L(Data|\lambda )\varphi (\lambda )}{\int_{0}^{\infty }L(Data|\lambda )\varphi (\lambda )d\lambda }</math>
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| where <math>\varphi (\lambda )=\tfrac{1}{\lambda }</math>, is the non-informative prior of <math>\lambda </math>.
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| With the above prior distribution, <math>f(\lambda |Data)</math> can be rewritten as:
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| ::<math>f(\lambda |Data)=\frac{L(Data|\lambda )\tfrac{1}{\lambda }}{\int_{0}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }</math>
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| The one-sided upper bound of <math>\lambda </math> is:
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| ::<math>CL=P(\lambda \le {{\lambda }_{U}})=\int_{0}^{{{\lambda }_{U}}}f(\lambda |Data)d\lambda </math>
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| The one-sided lower bound of <math>\lambda </math> is:
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| ::<math>1-CL=P(\lambda \le {{\lambda }_{L}})=\int_{0}^{{{\lambda }_{L}}}f(\lambda |Data)d\lambda </math>
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| The two-sided bounds of <math>\lambda </math> are:
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| ::<math>CL=P({{\lambda }_{L}}\le \lambda \le {{\lambda }_{U}})=\int_{{{\lambda }_{L}}}^{{{\lambda }_{U}}}f(\lambda |Data)d\lambda </math>
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| ====Bounds on Time (Type 1)====
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| The reliable life equation is:
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| ::<math>t=\frac{-\ln R}{\lambda }</math>
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| For the one-sided upper bound on time we have:
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| ::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(t\le {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln R}{\lambda }\le {{T}_{U}})</math>
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| The above equation can be rewritten in terms of <math>\lambda </math> as:
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| ::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln R}{{{t}_{U}}}\le \lambda )</math>
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| From the above posterior distribuiton equation, we have:
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| ::<math>CL=\frac{\int_{\tfrac{-\ln R}{{{t}_{U}}}}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }{\int_{0}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }</math>
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| The above equation is solved w.r.t. <math>{{t}_{U}}.</math> The same method is applied for one-sided lower and two-sided bounds on time.
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| ====Bounds on Reliability (Type 2)====
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| The one-sided upper bound on reliability is given by:
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| ::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\exp (-\lambda t)\le {{R}_{U}})</math>
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| The above equaation can be rewritten in terms of <math>\lambda </math> as:
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| ::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln {{R}_{U}}}{t}\le \lambda )</math>
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| From the equation for posterior distribution we have:
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| ::<math>CL=\frac{\int_{\tfrac{-\ln {{R}_{U}}}{t}}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }{\int_{0}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }</math>
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| The above equation is solved w.r.t. <math>{{R}_{U}}.</math> The same method can be used to calculate one-sided lower and two sided bounds on reliability.
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