Template:Bayesian Confidence Bounds ED: Difference between revisions

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

Latest revision as of 01:10, 13 August 2012

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