Template:Bayesian Confidence Bounds ED - Revision history

Richard House: Redirected page to The Exponential Distribution

2012-08-13T01:10:43Z

Redirected page to The Exponential Distribution

← Older revision		Revision as of 01:10, 13 August 2012
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	~~<br>~~		#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.~~

Harry Guo: /* Bounds on Parameters */

2012-02-10T23:03:31Z

Bounds on Parameters

← Older revision		Revision as of 23:03, 10 February 2012
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	===Bayesian Confidence Bounds===		===Bayesian Confidence Bounds===
	====Bounds on Parameters====		====Bounds on Parameters====
	From Chapter 5, we know that the posterior distribution of <math>\lambda </math> can be written as:		From Chapter [[Confidence Bounds]], we know that the posterior distribution of <math>\lambda </math> can be written as:

Harry Guo: /* Bayesian Confidence Bounds */

2012-02-08T22:05:04Z

Bayesian Confidence Bounds

← Older revision		Revision as of 22:05, 8 February 2012
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	<br>		<br>
	==Bayesian Confidence Bounds==		===Bayesian Confidence Bounds===
	===Bounds on Parameters===		====Bounds on Parameters====
	From Chapter 5, we know that the posterior distribution of <math>\lambda </math> can be written as:		From Chapter 5, we know that the posterior distribution of <math>\lambda </math> can be written as:

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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>		::<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)===		====Bounds on Time (Type 1)====
	The reliable life equation is:		The reliable life equation is:

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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.		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)===		====Bounds on Reliability (Type 2)====
	The one-sided upper bound on reliability is given by:		The one-sided upper bound on reliability is given by:

Harry Guo: /* Bounds on Reliability (Type 2) */

2012-02-08T01:14:23Z

Bounds on Reliability (Type 2)

← Older revision		Revision as of 01:14, 8 February 2012
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	The one-sided upper bound on reliability is given by:		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>		::<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:		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>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln {{R}_{U}}}{t}\le \lambda )</math>

Harry Guo: /* Bounds on Reliability (Type 2) */

2012-02-08T01:13:31Z

Bounds on Reliability (Type 2)

← Older revision		Revision as of 01:13, 8 February 2012
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	~~Eqn. (1SBR)~~ can be rewritten in terms of <math>\lambda </math> as:		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>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln {{R}_{U}}}{T}\le \lambda )</math>


	From ~~Eqn (postL),~~ we have:		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>		::<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>


	~~Eqn. (1CBR)~~ 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.		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.

Harry Guo: /* Bounds on Time (Type 1) */

2012-02-08T01:12:10Z

Bounds on Time (Type 1)

← Older revision		Revision as of 01:12, 8 February 2012
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	The above equation can be rewritten in terms of <math>\lambda </math> as:		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>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln R}{{{t}_{U}}}\le \lambda )</math>


	From the above posterior distribuiton equation, we have:		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>		::<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.		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)===		===Bounds on Reliability (Type 2)===

Harry Guo: /* Bounds on Time (Type 1) */

2012-02-08T00:42:57Z

Bounds on Time (Type 1)

← Older revision		Revision as of 00:42, 8 February 2012
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	::<math>T=\frac{-\ln R}{\lambda }</math>		::<math>t=\frac{-\ln R}{\lambda }</math>


	For the one-sided upper bound on time we have:		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>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(t\le {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln R}{\lambda }\le {{T}_{U}})</math>


	~~Eqn. (1SBT)~~ can be rewritten in terms of <math>\lambda </math> as:		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>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln R}{{{T}_{U}}}\le \lambda )</math>


	From ~~Eqn (postL)~~, we have:		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>		::<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>


	~~Eqn. (1CBT)~~ is solved w.r.t. <math>{{T}_{U}}.</math> The same method is applied for one-sided lower and two-sided bounds on time.		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)===		===Bounds on Reliability (Type 2)===

Harry Guo: /* Bayesian Confidence Bounds */

2012-02-08T00:33:41Z

Bayesian Confidence Bounds

← Older revision		Revision as of 00:33, 8 February 2012
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			<br>
	==Bayesian Confidence Bounds==		==Bayesian Confidence Bounds==
	===Bounds on Parameters===		===Bounds on Parameters===

Nicolette Young: Created page with '==Bayesian Confidence Bounds== ===Bounds on Parameters=== From Chapter 5, we know that the posterior distribution of $\lambda$ can be written as: :: $f(\lambda …'$

2012-01-04T16:19:36Z

Created page with '==Bayesian Confidence Bounds== ===Bounds on Parameters=== From Chapter 5, we know that the posterior distribution of <math>\lambda </math> can be written as: ::<math>f(\lambda …'

New page

==Bayesian Confidence Bounds==
===Bounds on Parameters===
From Chapter 5, 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>

Eqn. (1SBT) 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 Eqn (postL), 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>

Eqn. (1CBT) 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>

Eqn. (1SBR) 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 Eqn (postL), 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>

Eqn. (1CBR) 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.

Template:Bayesian Confidence Bounds ED - Revision history

Richard House: Redirected page to The Exponential Distribution

Harry Guo: /* Bounds on Parameters */

Harry Guo: /* Bayesian Confidence Bounds */

Harry Guo: /* Bounds on Reliability (Type 2) */

Harry Guo: /* Bounds on Reliability (Type 2) */

Harry Guo: /* Bounds on Time (Type 1) */

Harry Guo: /* Bounds on Time (Type 1) */

Harry Guo: /* Bayesian Confidence Bounds */

Nicolette Young: Created page with '==Bayesian Confidence Bounds== ===Bounds on Parameters=== From Chapter 5, we know that the posterior distribution of \lambda can be written as: ::f(\lambda …'

Nicolette Young: Created page with '==Bayesian Confidence Bounds== ===Bounds on Parameters=== From Chapter 5, we know that the posterior distribution of $\lambda$ can be written as: :: $f(\lambda …'$