Template:Eyring-log stat prop sum - Revision history

Richard House: Redirected page to Eyring Relationship#Eyring-Lognormal

2012-08-16T23:32:47Z

Redirected page to Eyring Relationship#Eyring-Lognormal

← Older revision		Revision as of 23:32, 16 August 2012
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	~~===~~Eyring-Lognormal ~~Statistical Properties Summary===~~		#REDIRECT [[Eyring_Relationship#Eyring-Lognormal]]

	~~{{eyring-log mean}}~~

	~~{{eyring-log median}}~~

	~~{{eyring-log sd}}~~

	~~{{eyring-log mode}}~~

	~~{{eyring-log rf}}~~

	~~{{eyring-log rl}}~~

	~~{{eyring-log fr}}~~

Nicolette Young: /* Eyring-Lognormal Failure Rate */

2012-01-12T23:06:13Z

Eyring-Lognormal Failure Rate

← Older revision		Revision as of 23:06, 12 January 2012
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	{{eyring-log rl}}		{{eyring-log rl}}

	~~====Eyring-Lognormal Failure Rate====~~		{{eyring-log fr}}

	~~<br>~~
	~~The Eyring-lognormal failure rate is given by:~~

	~~::<math>\lambda (T,V)=\frac{f(T,V)}{R(T,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{~~{~~e}^~~{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\mathop{}_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}~~</math>~~

Nicolette Young: /* Reliable Life */

2012-01-12T23:05:44Z

Reliable Life

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 {{eyring-log rf}}
-====Reliable Life====
+{{eyring-log rl}}
 ====Eyring-Lognormal Failure Rate====

Nicolette Young: /* Eyring-Lognormal Reliability Function */

2012-01-12T23:05:14Z

Eyring-Lognormal Reliability Function

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 {{eyring-log mode}}
-====Eyring-Lognormal Reliability Function====
+{{eyring-log rf}}
 ====Reliable Life====

Nicolette Young: /* The Mode */

2012-01-12T23:04:54Z

The Mode

← Older revision		Revision as of 23:04, 12 January 2012
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	{{eyring-log sd}}		{{eyring-log sd}}

	~~====The Mode====~~		{{eyring-log mode}}
	~~<br>~~
	~~• The mode of the Eyring-lognormal model is given by:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& \tilde{T}= & {{e}^{{{\overline~~{~~T}}^~~{~~\prime }}~~-~~\sigma _{{{T}'}}^{2}}} \\~~
	~~& = & {{e}^{-\ln (V)-A+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}~~}}
	~~\end{align}</math>~~

	====Eyring-Lognormal Reliability Function====		====Eyring-Lognormal Reliability Function====

Nicolette Young: /* The Standard Deviation */

2012-01-12T23:04:22Z

The Standard Deviation

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 {{eyring-log median}}
-====The Standard Deviation====
+{{eyring-log sd}}
 ====The Mode====

Nicolette Young: /* The Median */

2012-01-12T23:04:03Z

The Median

← Older revision		Revision as of 23:04, 12 January 2012
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	{{eyring-log mean}}		{{eyring-log mean}}

	~~====The Median====~~		{{eyring-log median}}
	~~<br>~~
	~~The median of the Eyring-lognormal model is given by:~~

	~~<br>~~
	~~::<math>\breve~~{~~T}=~~{~~{e}^{{{\overline{T}}^{\prime }}~~}}~~</math>~~

	~~<br>~~

	====The Standard Deviation====		====The Standard Deviation====

Nicolette Young: /* The Mean */

2012-01-12T23:03:46Z

The Mean

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 ===Eyring-Lognormal Statistical Properties Summary===
-====The Mean====
+{{eyring-log mean}}
 ====The Median====

Nicolette Young: Created page with '===Eyring-Lognormal Statistical Properties Summary=== ====The Mean====
• The mean life of the Eyring-lognormal model (mean of the times-to-failure), $\bar{T}$ …'

2012-01-12T23:01:08Z

Created page with '===Eyring-Lognormal Statistical Properties Summary=== ====The Mean==== • The mean life of the Eyring-lognormal model (mean of the times-to-failure), <math>\bar{T}</math>…'

New page

===Eyring-Lognormal Statistical Properties Summary===

====The Mean====
 
• The mean life of the Eyring-lognormal model (mean of the times-to-failure), <math>\bar{T}</math> , is given by:
 

::<math>\begin{align}
& \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} \\
& = & {{e}^{-\ln (V)-A+\tfrac{B}{V}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}
\end{align}</math>

 
The mean of the natural logarithms of the times-to-failure, <math>{{\bar{T}}^{^{\prime }}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:
 

 
::<math>{{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)</math>
 

====The Median====
 
The median of the Eyring-lognormal model is given by:

 
::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>

 

====The Standard Deviation====
 
• The standard deviation of the Eyring-lognormal model (standard deviation of the times-to-failure), <math>{{\sigma }_{T}}</math> , is given by:

 
::<math>\begin{align}
& {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \\
& = & \sqrt{\left( {{e}^{2\left( -\ln (V)-A+\tfrac{B}{V} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}
\end{align}</math>

 
• The standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:

 
::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}</math>

====The Mode====
 
• The mode of the Eyring-lognormal model is given by:

 
::<math>\begin{align}
& \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} \\
& = & {{e}^{-\ln (V)-A+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}}
\end{align}</math>

====Eyring-Lognormal Reliability Function====
 
The reliability for a mission of time <math>T</math> , starting at age 0, for the Eyring-lognormal model is determined by:

 
::<math>R(T,V)=\mathop{}_{T}^{\infty }f(t,V)dt</math>

 
:or:

 
::<math>R(T,V)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>

 
There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods.

====Reliable Life====
 
For the Eyring-lognormal model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},</math> is estimated by first solving the reliability equation with respect to time, as follows:

 
::<math>T_{R}^{\prime }=-\ln (V)-A+\frac{B}{V}+z\cdot {{\sigma }_{{{T}'}}}</math>

 
:where:

 
::<math>z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },V \right) \right]</math>

 
:and:

 
::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}',V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>
 
 
Since <math>{T}'=\ln (T)</math> the reliable life, <math>{{t}_{R,}}</math> is given by:

 
::<math>{{t}_{R}}={{e}^{T_{R}^{\prime }}}</math>

====Eyring-Lognormal Failure Rate====

 
The Eyring-lognormal failure rate is given by:

::<math>\lambda (T,V)=\frac{f(T,V)}{R(T,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\mathop{}_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (V)+A-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>