Template:Alta al stat prop sum - Revision history

Richard House: Redirected page to Arrhenius Relationship#Arrhenius-Lognormal Statistical Properties Summary

2012-08-16T05:30:33Z

Redirected page to Arrhenius Relationship#Arrhenius-Lognormal Statistical Properties Summary

← Older revision		Revision as of 05:30, 16 August 2012
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	~~===~~Arrhenius-~~Lognormal Statistical Properties Summary===~~		#REDIRECT [[Arrhenius_Relationship#Arrhenius-Lognormal_Statistical_Properties_Summary]]
	~~{{alta al mean}}~~

	~~{{alta al sd}}~~

	~~{{alta al mode}}~~

	~~{{alta al rf}}~~

	~~{{alta al rl}}~~

	~~{{alta al fr}}~~

	~~<br>~~
	~~<br>~~

Athanasios Gerokostopoulos at 00:45, 14 February 2012

2012-02-14T00:45:46Z

← Older revision		Revision as of 00:45, 14 February 2012
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	{{alta al fr}}		{{alta al fr}}

	~~====When Using the Lognormal Distribution in ALTA====~~
	~~<br>~~
	The parameters returned for the Arrhenius-lognormal distribution are always <math>{{\sigma }_{{{T}'}}},</math> <math>C,</math> and <math>B.</math> The returned <math>{{\sigma }_{{{T}'}}}</math> is always the square root of the variance of the natural logarithms to failure. Also, if the "Show Scale Parameter" option is checked (on the Data Sheet tab in the User Setup), the returned mean value is always the mean of the natural logarithms of the times-to-failure, given by Eqn. (arrh-logn-mean). Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as parameters of the distribution, and are thus the mean (a function of stress as it can be seen in Eqn. (arrh-logn-mean)) and standard deviation of the natural logarithms of the data. The mean life value of the times-to-failure, as well as the standard deviation of times-to-failure (not the parameter) can be obtained through the Function Wizard in ALTA.
	<br>		<br>
	<br>		<br>

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

2012-01-12T22:12:03Z

Arrhenius-Lognormal Failure Rate

← Older revision		Revision as of 22:12, 12 January 2012
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	{{alta al rl}}		{{alta al rl}}

	~~====Arrhenius-Lognormal Failure Rate====~~		{{alta al fr}}
	~~<br>~~
	~~The Arrhenius-lognormal failure rate is given by:~~
	~~<br>~~
	::<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 (C)-\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 (C)-\tfrac{B}{V}}{{{\sigma }_{{{~~T}'}}}} \right)}^{2}}}~~}dt}~~</math>~~

	====When Using the Lognormal Distribution in ALTA====		====When Using the Lognormal Distribution in ALTA====

Nicolette Young: /* Reliable Life */

2012-01-12T22:11:07Z

Reliable Life

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 {{alta al rf}}
-====Reliable Life====
+{{alta al rl}}
 ====Arrhenius-Lognormal Failure Rate====

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

2012-01-12T22:10:45Z

Arrhenius-Lognormal Reliability Function

← Older revision		Revision as of 22:10, 12 January 2012
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	{{alta al mode}}		{{alta al mode}}

	~~====Arrhenius-Lognormal Reliability Function====~~		{{alta al rf}}
	~~<br>~~
	~~The reliability for a mission of time <math>T</math> , starting at age 0, for the Arrhenius-lognormal model is determined by:~~
	~~<br>~~
	~~::<math>R(T,V)=\mathop~~{}_{T}~~^{\infty~~ }~~f(t,V)dt</math>~~

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

Nicolette Young: /* The Mode */

2012-01-12T22:10:23Z

The Mode

← Older revision		Revision as of 22:10, 12 January 2012
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	{{alta al sd}}		{{alta al sd}}

	~~====The Mode====~~		{{alta al mode}}
	~~<br>~~
	~~• The mode of the Arrhenius-lognormal model is given by:~~

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


	====Arrhenius-Lognormal Reliability Function====		====Arrhenius-Lognormal Reliability Function====
	<br>		<br>

Nicolette Young: /* The Standard Deviation */

2012-01-12T22:10:01Z

The Standard Deviation

← Older revision		Revision as of 22:10, 12 January 2012
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	{{alta al mean}}		{{alta al mean}}

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

	~~<br>~~
	~~::<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 (C)+\tfrac{B}{V} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}~~
	~~\end{align}</math>~~

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

Nicolette Young: /* The Mean */

2012-01-12T22:09:37Z

The Mean

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 ===Arrhenius-Lognormal Statistical Properties Summary===
-====The Mean====
+{{alta al mean}}
 ====The Standard Deviation====
 <br>

Nicolette Young: Created page with '===Arrhenius-Lognormal Statistical Properties Summary=== ====The Mean====
• The mean life of the Arrhenius-lognormal model (mean of the times-to-failure), $\bar{T}$
2012-01-12T21:50:00Z

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

New page
===Arrhenius-Lognormal Statistical Properties Summary===
====The Mean====
 
• The mean life of the Arrhenius-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 (C)+\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 Standard Deviation====
 
• The standard deviation of the Arrhenius-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 (C)+\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 Arrhenius-lognormal model is given by:

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

====Arrhenius-Lognormal Reliability Function====
 
The reliability for a mission of time <math>T</math> , starting at age 0, for the Arrhenius-lognormal model is determined by:
 
::<math>R(T,V)=\mathop{}_{T}^{\infty }f(t,V)dt</math>

vor:
 
::<math>R(T,V)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (C)-\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 Arrhenius-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,
..
 
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>

====Arrhenius-Lognormal Failure Rate====
 
The Arrhenius-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 (C)-\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 (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>

====When Using the Lognormal Distribution in ALTA====
 
The parameters returned for the Arrhenius-lognormal distribution are always <math>{{\sigma }_{{{T}'}}},</math> <math>C,</math> and <math>B.</math> The returned <math>{{\sigma }_{{{T}'}}}</math> is always the square root of the variance of the natural logarithms to failure. Also, if the "Show Scale Parameter" option is checked (on the Data Sheet tab in the User Setup), the returned mean value is always the mean of the natural logarithms of the times-to-failure, given by Eqn. (arrh-logn-mean). Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as parameters of the distribution, and are thus the mean (a function of stress as it can be seen in Eqn. (arrh-logn-mean)) and standard deviation of the natural logarithms of the data. The mean life value of the times-to-failure, as well as the standard deviation of times-to-failure (not the parameter) can be obtained through the Function Wizard in ALTA.