Template:Lognormal distribution bayesian confidence bounds - Revision history

Richard House: Redirected page to The Lognormal Distribution#Bayesian Confidence Bounds

2012-08-13T06:09:25Z

Redirected page to The Lognormal Distribution#Bayesian Confidence Bounds

← Older revision		Revision as of 06:09, 13 August 2012
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	~~===Bayesian Confidence Bounds===~~		#REDIRECT [[The_Lognormal_Distribution#Bayesian_Confidence_Bounds]]
	~~====Bounds on Parameters====~~
	~~From Chapter~~ [[~~Parameter Estimation~~]]~~, we know that the marginal distribution of parameter <math>{\mu }'</math> is:~~

	~~::<math>\begin{align}~~
	~~f({\mu }'\|Data)= & \int_{0}^{\infty }f({\mu }',{{\sigma'}}\|Data)d{{\sigma'}} \\~~
	= & \frac{\int_{0}^{\infty }L(Data\|{\mu }',{{\sigma'}})\varphi ({\mu }')\varphi ({{\sigma'}})d{{\sigma'}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|{\mu }',{{\sigma'}})\varphi ({\mu }')\varphi ({{\sigma'}})d{\mu }'d{{\sigma'}}}
	~~\end{align}</math>~~

	~~where:~~
	~~::<math>\varphi ({{\sigma '}})</math> is <math>\tfrac{1}{{{\sigma '}}}</math> , non-informative prior of <math>{{\sigma '}}</math> .~~
	~~<math>\varphi ({\mu }')</math> is an uniform distribution from - <math>\infty </math> to + <math>\infty </math> , non-informative prior of <math>{\mu }'</math> .~~
	~~With the above prior distributions, <math>f({\mu }'\|Data)</math> can be rewritten as:~~


	::<math>f({\mu }'\|Data)=\frac{\int_{0}^{\infty }L(Data\|{\mu }',{{\sigma '}})\tfrac{1}{{{\sigma '}}}d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|{\mu }',{{\sigma '}})\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}</math>


	~~The one-sided upper bound of <math>{\mu }'</math> is:~~


	~~::<math>CL=P({\mu }'\le \mu _{U}^{\prime })=\int_{-\infty }^{\mu _{U}^{\prime }}f({\mu }'\|Data)d{\mu }'</math>~~


	~~The one-sided lower bound of <math>{\mu }'</math> is:~~


	~~::<math>1-CL=P({\mu }'\le \mu _{L}^{\prime })=\int_{-\infty }^{\mu _{L}^{\prime }}f({\mu }'\|Data)d{\mu }'</math>~~


	~~The two-sided bounds of <math>{\mu }'</math> is:~~


	~~::<math>CL=P(\mu _{L}^{\prime }\le {\mu }'\le \mu _{U}^{\prime })=\int_{\mu _{L}^{\prime }}^{\mu _{U}^{\prime }}f({\mu }'\|Data)d{\mu }'</math>~~


	~~The same method can be used to obtained the bounds of <math>{{\sigma '}}</math> .~~

	~~====Bounds on Time (Type 1)====~~
	~~The reliable life of the lognormal distribution is:~~


	~~::<math>\ln T={\mu }'+{{\sigma '}}{{\Phi }^{-1}}(1-R)</math>~~


	~~The one-sided upper on time bound is given by:~~


	~~::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\ln t\le \ln {{t}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'+{{\sigma '}}{{\Phi }^{-1}}(1-R)\le \ln {{t}_{U}})</math>~~


	~~The above equation can be rewritten in terms of <math>{\mu }'</math> as:~~


	~~::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{t}_{U}}-{{\sigma '}}{{\Phi }^{-1}}(1-R)</math>~~


	~~From the posterior distribution of <math>{\mu }'</math> get:~~


	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln {{t}_{U}}-{{\sigma ‘}}{{\Phi }^{-1}}(1-R)}L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}</math>


	~~The above equation is solved w.r.t. <math>{{t}_{U}}.</math> The same method can be applied for one-sided lower bounds 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 }}}\,({\mu }'\le \ln t-{{\sigma '}}{{\Phi }^{-1}}(1-{{R}_{U}}))</math>~~


	~~From the posterior distribution of <math>{\mu }'</math> is:~~


	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln t-{{\sigma '}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma'}},{\mu }')\tfrac{1}{{{\sigma'}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}</math>


	~~The above equation is solved w.r.t. <math>{{R}_{U}}.</math> The same method is used to calculate the one-sided lower bounds and two-sided bounds on Reliability.~~


	~~'''Example 8:'''~~
	~~{{Example: Lognormal Distribution Bayesian Bound (Parameters)}}~~

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

2012-02-13T23:40:40Z

Bounds on Reliability (Type 2)

← Older revision		Revision as of 23:40, 13 February 2012
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	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln t-{{\sigma 1}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma'}},{\mu }')\tfrac{1}{{{\sigma'}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}</math>		::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln t-{{\sigma '}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma'}},{\mu }')\tfrac{1}{{{\sigma'}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}</math>

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

2012-02-13T23:40:18Z

Bounds on Reliability (Type 2)

← Older revision		Revision as of 23:40, 13 February 2012
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	::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln T-{{\sigma '}}{{\Phi }^{-1}}(1-{{R}_{U}}))</math>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln t-{{\sigma '}}{{\Phi }^{-1}}(1-{{R}_{U}}))</math>


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	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln T-{{\sigma 1}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma'}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma '}}}</math>		::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln t-{{\sigma 1}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma'}},{\mu }')\tfrac{1}{{{\sigma'}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}</math>


	~~Eqn. (1SCBR)~~ is solved w.r.t. <math>{{R}_{U}}.</math> The same method is used to calculate the one-sided lower bounds and two-sided bounds on Reliability.		The above equation is solved w.r.t. <math>{{R}_{U}}.</math> The same method is used to calculate the one-sided lower bounds and two-sided bounds on Reliability.


	'''Example 8:'''		'''Example 8:'''
	{{Example: Lognormal Distribution Bayesian Bound (Parameters)}}		{{Example: Lognormal Distribution Bayesian Bound (Parameters)}}

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

2012-02-13T23:39:06Z

Bounds on Time (Type 1)

← Older revision		Revision as of 23:39, 13 February 2012
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	::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\ln T\le \ln {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'+{{\sigma '}}{{\Phi }^{-1}}(1-R)\le \ln {{T}_{U}})</math>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\ln t\le \ln {{t}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'+{{\sigma '}}{{\Phi }^{-1}}(1-R)\le \ln {{t}_{U}})</math>


	~~Eqn. (1SBT)~~ can be rewritten in terms of <math>{\mu }'</math> as:		The above equation can be rewritten in terms of <math>{\mu }'</math> as:


	::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{T}_{U}}-{{\sigma '}}{{\Phi }^{-1}}(1-R)</math>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{t}_{U}}-{{\sigma '}}{{\Phi }^{-1}}(1-R)</math>


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	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln {{T}_{U}}-{{\sigma ‘}}{{\Phi }^{-1}}(1-R)}L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}</math>		::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln {{t}_{U}}-{{\sigma ‘}}{{\Phi }^{-1}}(1-R)}L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}</math>


	~~Eqn. (1SCBT)~~ is solved w.r.t. <math>{{T}_{U}}.</math> The same method can be applied for one-sided lower bounds and two-sided bounds on Time.		The above equation is solved w.r.t. <math>{{t}_{U}}.</math> The same method can be applied for one-sided lower bounds and two-sided bounds on Time.


	====Bounds on Reliability (Type 2)====		====Bounds on Reliability (Type 2)====

Harry Guo: /* Bayesian Confidence Bounds */

2012-02-13T23:37:05Z

Bayesian Confidence Bounds

← Older revision		Revision as of 23:37, 13 February 2012
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	\end{align}</math>		\end{align}</math>

	:where:		where:
	::<math>\varphi ({{\sigma '}})</math> is <math>\tfrac{1}{{{\sigma '}}}</math> , non-informative prior of <math>{{\sigma '}}</math> .		::<math>\varphi ({{\sigma '}})</math> is <math>\tfrac{1}{{{\sigma '}}}</math> , non-informative prior of <math>{{\sigma '}}</math> .
	<math>\varphi ({\mu }')</math> is an uniform distribution from - <math>\infty </math> to + <math>\infty </math> , non-informative prior of <math>{\mu }'</math> .		<math>\varphi ({\mu }')</math> is an uniform distribution from - <math>\infty </math> to + <math>\infty </math> , non-informative prior of <math>{\mu }'</math> .

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

2012-02-13T23:36:28Z

Bounds on Reliability (Type 2)

← Older revision		Revision as of 23:36, 13 February 2012
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	Eqn. (1SCBR) is solved w.r.t. <math>{{R}_{U}}.</math> The same method is used to calculate the one-sided lower bounds and two-sided bounds on Reliability.		Eqn. (1SCBR) is solved w.r.t. <math>{{R}_{U}}.</math> The same method is used to calculate the one-sided lower bounds and two-sided bounds on Reliability.


	'''Example 8:'''		'''Example 8:'''
	{{Example: Lognormal Distribution Bayesian Bound (Parameters)}}		{{Example: Lognormal Distribution Bayesian Bound (Parameters)}}

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

2012-02-13T23:36:14Z

Bounds on Reliability (Type 2)

← Older revision		Revision as of 23:36, 13 February 2012
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	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln T-{{\sigma 1}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}</math>		::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln T-{{\sigma 1}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma'}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma '}}}</math>


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	'''Example 8:'''		'''Example 8:'''
	{{Example: Lognormal ~~Distr~~		{{Example: Lognormal Distribution Bayesian Bound (Parameters)}}

Harry Guo: /* Bayesian Confidence Bounds */

2012-02-13T23:35:06Z

Bayesian Confidence Bounds

← Older revision		Revision as of 23:35, 13 February 2012
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	:where:		:where:
	::<math>\varphi ({{\sigma ‘}})</math> is <math>\tfrac{1}{{{\sigma ‘}}}</math> , non-informative prior of <math>{{\sigma ‘}}</math> .		::<math>\varphi ({{\sigma '}})</math> is <math>\tfrac{1}{{{\sigma '}}}</math> , non-informative prior of <math>{{\sigma '}}</math> .
	<math>\varphi ({\mu }')</math> is an uniform distribution from - <math>\infty </math> to + <math>\infty </math> , non-informative prior of <math>{\mu }'</math> .		<math>\varphi ({\mu }')</math> is an uniform distribution from - <math>\infty </math> to + <math>\infty </math> , non-informative prior of <math>{\mu }'</math> .
	With the above prior distributions, <math>f({\mu }'\|Data)</math> can be rewritten as:		With the above prior distributions, <math>f({\mu }'\|Data)</math> can be rewritten as:


	::<math>f({\mu }'\|Data)=\frac{\int_{0}^{\infty }L(Data\|{\mu }',{{\sigma ‘}})\tfrac{1}{{{\sigma ‘}}}d{{\sigma ‘}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|{\mu }',{{\sigma ‘}})\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}</math>		::<math>f({\mu }'\|Data)=\frac{\int_{0}^{\infty }L(Data\|{\mu }',{{\sigma '}})\tfrac{1}{{{\sigma '}}}d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|{\mu }',{{\sigma '}})\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}</math>


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	The same method can be used to obtained the bounds of <math>{{\sigma ‘}}</math> .		The same method can be used to obtained the bounds of <math>{{\sigma '}}</math> .

	====Bounds on Time (Type 1)====		====Bounds on Time (Type 1)====
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	::<math>\ln T={\mu }'+{{\sigma ‘}}{{\Phi }^{-1}}(1-R)</math>		::<math>\ln T={\mu }'+{{\sigma '}}{{\Phi }^{-1}}(1-R)</math>


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	::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\ln T\le \ln {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'+{{\sigma ‘}}{{\Phi }^{-1}}(1-R)\le \ln {{T}_{U}})</math>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\ln T\le \ln {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'+{{\sigma '}}{{\Phi }^{-1}}(1-R)\le \ln {{T}_{U}})</math>


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	::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{T}_{U}}-{{\sigma ‘}}{{\Phi }^{-1}}(1-R)</math>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{T}_{U}}-{{\sigma '}}{{\Phi }^{-1}}(1-R)</math>


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	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln {{T}_{U}}-{{\sigma ‘}}{{\Phi }^{-1}}(1-R)}L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}</math>		::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln {{T}_{U}}-{{\sigma ‘}}{{\Phi }^{-1}}(1-R)}L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}</math>


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	::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln T-{{\sigma ‘}}{{\Phi }^{-1}}(1-{{R}_{U}}))</math>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln T-{{\sigma '}}{{\Phi }^{-1}}(1-{{R}_{U}}))</math>


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	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln T-{{\sigma ‘}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}</math>		::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln T-{{\sigma 1}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}</math>

Harry Guo: /* Bayesian Confidence Bounds */

2012-02-13T23:33:05Z

Bayesian Confidence Bounds

← Older revision		Revision as of 23:33, 13 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	f({\mu }'\|Data)= & \int_{0}^{\infty }f({\mu }',{{\sigma'}}\|Data)d{{\sigma'}} \\		f({\mu }'\|Data)= & \int_{0}^{\infty }f({\mu }',{{\sigma'}}\|Data)d{{\sigma'}} \\
	= & \frac{\int_{0}^{\infty }L(Data\|{\mu }',{{\sigma ‘}})\varphi ({\mu }')\varphi ({{\sigma ‘}})d{{\sigma ‘}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|{\mu }',{{\sigma ‘}})\varphi ({\mu }')\varphi ({{\sigma ‘}})d{\mu }'d{{\sigma ‘}}}		= & \frac{\int_{0}^{\infty }L(Data\|{\mu }',{{\sigma'}})\varphi ({\mu }')\varphi ({{\sigma'}})d{{\sigma'}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|{\mu }',{{\sigma'}})\varphi ({\mu }')\varphi ({{\sigma'}})d{\mu }'d{{\sigma'}}}
	\end{align}</math>		\end{align}</math>

Harry Guo at 23:31, 13 February 2012

2012-02-13T23:31:36Z

← Older revision		Revision as of 23:31, 13 February 2012
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	:where:		:where:
	::<math>\varphi ({{\sigma ~~}_{{{T}'}~~}})</math> is <math>\tfrac{1}{{{\sigma ~~}_{{{T}'}~~}}}</math> , non-informative prior of <math>{{\sigma ~~}_{{{T}'}~~}}</math> .		::<math>\varphi ({{\sigma ‘}})</math> is <math>\tfrac{1}{{{\sigma ‘}}}</math> , non-informative prior of <math>{{\sigma ‘}}</math> .
	<math>\varphi ({\mu }')</math> is an uniform distribution from - <math>\infty </math> to + <math>\infty </math> , non-informative prior of <math>{\mu }'</math> .		<math>\varphi ({\mu }')</math> is an uniform distribution from - <math>\infty </math> to + <math>\infty </math> , non-informative prior of <math>{\mu }'</math> .
	With the above prior distributions, <math>f({\mu }'\|Data)</math> can be rewritten as:		With the above prior distributions, <math>f({\mu }'\|Data)</math> can be rewritten as:


	::<math>f({\mu }'\|Data)=\frac{\int_{0}^{\infty }L(Data\|{\mu }',{{\sigma ~~}_{{{T}'}~~}})\tfrac{1}{{{\sigma ~~}_{{{T}'}~~}}}d{{\sigma ~~}_{{{T}'}~~}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|{\mu }',{{\sigma ~~}_{{{T}'}~~}})\tfrac{1}{{{\sigma ~~}_{{{T}'}~~}}}d{\mu }'d{{\sigma ~~}_{{{T}'}~~}}}</math>		::<math>f({\mu }'\|Data)=\frac{\int_{0}^{\infty }L(Data\|{\mu }',{{\sigma ‘}})\tfrac{1}{{{\sigma ‘}}}d{{\sigma ‘}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|{\mu }',{{\sigma ‘}})\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}</math>


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	The same method can be used to obtained the bounds of <math>{{\sigma ~~}_{{{T}'}~~}}</math> .		The same method can be used to obtained the bounds of <math>{{\sigma ‘}}</math> .

	====Bounds on Time (Type 1)====		====Bounds on Time (Type 1)====
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	::<math>\ln T={\mu }'+{{\sigma ~~}_{{{T}'}~~}}{{\Phi }^{-1}}(1-R)</math>		::<math>\ln T={\mu }'+{{\sigma ‘}}{{\Phi }^{-1}}(1-R)</math>


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	::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\ln T\le \ln {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'+{{\sigma ~~}_{{{T}'}~~}}{{\Phi }^{-1}}(1-R)\le \ln {{T}_{U}})</math>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\ln T\le \ln {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'+{{\sigma ‘}}{{\Phi }^{-1}}(1-R)\le \ln {{T}_{U}})</math>


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	::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{T}_{U}}-{{\sigma ~~}_{{{T}'}~~}}{{\Phi }^{-1}}(1-R)</math>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{T}_{U}}-{{\sigma ‘}}{{\Phi }^{-1}}(1-R)</math>


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	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln {{T}_{U}}-{{\sigma ~~}_{{{T}'}~~}}{{\Phi }^{-1}}(1-R)}L({{\sigma ~~}_{{{T}'}~~}},{\mu }')\tfrac{1}{{{\sigma ~~}_{{{T}'}~~}}}d{\mu }'d{{\sigma ~~}_{{{T}'}~~}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma ~~}_{{{T}'}~~}},{\mu }')\tfrac{1}{{{\sigma ~~}_{{{T}'}~~}}}d{\mu }'d{{\sigma ~~}_{{{T}'}~~}}}</math>		::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln {{T}_{U}}-{{\sigma ‘}}{{\Phi }^{-1}}(1-R)}L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}</math>


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	::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln T-{{\sigma ~~}_{{{T}'}~~}}{{\Phi }^{-1}}(1-{{R}_{U}}))</math>		::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln T-{{\sigma ‘}}{{\Phi }^{-1}}(1-{{R}_{U}}))</math>


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	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln T-{{\sigma ~~}_{{{T}'}~~}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma ~~}_{{{T}'}~~}},{\mu }')\tfrac{1}{{{\sigma ~~}_{{{T}'}~~}}}d{\mu }'d{{\sigma ~~}_{{{T}'}~~}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma ~~}_{{{T}'}~~}},{\mu }')\tfrac{1}{{{\sigma ~~}_{{{T}'}~~}}}d{\mu }'d{{\sigma ~~}_{{{T}'}~~}}}</math>		::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln T-{{\sigma ‘}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}</math>


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	'''Example 8:'''		'''Example 8:'''
	{{Example: Lognormal ~~Distribution Bayesian Bound (Parameters)}}~~		{{Example: Lognormal Distr