Template:Aae stat prop sum - Revision history

Lisa Hacker: Redirected page to Arrhenius Relationship#Arrhenius-Exponential

2012-08-16T00:06:02Z

Redirected page to Arrhenius Relationship#Arrhenius-Exponential

← Older revision		Revision as of 00:06, 16 August 2012
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	~~===~~Arrhenius-Exponential ~~Statistical Properties Summary===~~		#REDIRECT [[Arrhenius_Relationship#Arrhenius-Exponential]]
	~~<br>~~
	~~{{aae mean}}~~
	~~{{aae median}}~~

	~~{{aae mode}}~~

	~~{{aae sd}}~~

	~~{{aae rf}}~~

	~~{{aae cr}}~~

	~~{{aae rl}}~~

Nicolette Young: /* Reliable Life */

2012-01-12T19:03:33Z

Reliable Life

← Older revision		Revision as of 19:03, 12 January 2012
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	{{aae cr}}		{{aae cr}}

	~~====Reliable Life====~~		{{aae rl}}
	~~<br>~~
	~~For the Arrhenius-exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>~~{{~~t}_{R}},</math> is given by:~~

	~~<br>~~
	~~::<math>R({{t}_{R}},V)={{e}^{-\tfrac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}}}</math>~~



	~~<br>~~
	~~::<math>\ln [R({{t}_{R}},V)]=-\frac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}</math>~~

	~~<br>~~
	~~or:~~

	~~<br>~~
	~~::<math>{{t}_{R}}=-C{{e}^{\tfrac{B}{V}}}\ln [R({{t}_{R~~}}~~,V)]</math>~~

Nicolette Young: /* Conditional Reliability */

2012-01-12T19:03:14Z

Conditional Reliability

← Older revision		Revision as of 19:03, 12 January 2012
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	{{aae rf}}		{{aae rf}}

	~~====Conditional Reliability====~~		{{aae cr}}
	~~The Arrhenius-exponential conditional reliability function is given by,~~

	~~::<math>R(T,t,V)=\frac~~{~~R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{C{{e}^~~{~~\tfrac{B}{V}}}}~~}}~~</math>~~

	====Reliable Life====		====Reliable Life====

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

2012-01-12T19:02:57Z

Arrhenius-Exponential Reliability Function

@@ Line 8: / Line 8: @@
 {{aae sd}}
-====Arrhenius-Exponential Reliability Function====
+{{aae rf}}
 ====Conditional Reliability====

Nicolette Young: /* Standard Deviation */

2012-01-12T19:02:35Z

Standard Deviation

← Older revision		Revision as of 19:02, 12 January 2012
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	{{aae mode}}		{{aae mode}}

	~~====Standard Deviation====~~		{{aae sd}}
	~~The standard deviation, <math>~~{{~~\sigma }_{T}}</math> , of the Arrhenius-exponential model is given by:~~

	~~<br>~~
	~~::<math>{{\sigma }_{T}}=C{{e}^{\tfrac{B}{V}~~}}~~</math>~~

	====Arrhenius-Exponential Reliability Function====		====Arrhenius-Exponential Reliability Function====

Nicolette Young: /* Mode */

2012-01-12T19:02:07Z

Mode

← Older revision		Revision as of 19:02, 12 January 2012
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	{{aae median}}		{{aae median}}

	~~====Mode====~~		{{aae mode}}
	~~<br>~~
	~~The~~ mode~~, <math>\tilde{T~~}~~,</math>~~
	~~of the Arrhenius-exponential model is given by:~~

	~~<br>~~
	~~::<math>\tilde{T~~}~~=0</math>~~


	~~<br>~~

	====Standard Deviation====		====Standard Deviation====

Nicolette Young: /* Median */

2012-01-12T19:01:51Z

Median

← Older revision		Revision as of 19:01, 12 January 2012
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	<br>		<br>
	{{aae mean}}		{{aae mean}}
	~~====Median====~~		{{aae median}}
	~~<br>~~
	~~The median, <math>breve~~{~~T},</math>~~
	~~of the Arrhenius-exponential model is given by:~~

	~~<br>~~
	~~::<math>\breve~~{~~T}=0.693\cdot C{{e}^{\tfrac{B}{V}~~}}~~</math>~~


	~~<br>~~

	====Mode====		====Mode====

Nicolette Young: /* Arrhenius-Exponential Statistical Properties Summary */

2012-01-12T19:00:55Z

Arrhenius-Exponential Statistical Properties Summary

← Older revision		Revision as of 19:00, 12 January 2012
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	===Arrhenius-Exponential Statistical Properties Summary===		===Arrhenius-Exponential Statistical Properties Summary===
	<br>		<br>
	~~====Mean or MTTF====~~		{{aae mean}}
	~~<br>~~
	~~The mean, <math>\overline~~{~~T},</math> or Mean Time To Failure (MTTF) of the Arrhenius-exponential is given by,~~


	~~<br>~~
	~~::<math>\begin~~{~~align}~~
	~~& \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t,V)dt=\mathop{}_{0}^{\infty }t\cdot \frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V~~}}~~}}}}dt \\~~
	~~& = & C{{e}^{\tfrac{B}{V}}}~~
	~~\end{align}</math>~~

	~~<br>~~
	====Median====		====Median====
	<br>		<br>

Nicolette Young: Created page with '===Arrhenius-Exponential Statistical Properties Summary===
====Mean or MTTF====
The mean, $\overline{T},$ or Mean Time To Failure (MTTF) of the Arrhenius-e…'

2012-01-12T18:59:54Z

Created page with '===Arrhenius-Exponential Statistical Properties Summary=== ====Mean or MTTF==== The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) of the Arrhenius-e…'

New page

===Arrhenius-Exponential Statistical Properties Summary===
 
====Mean or MTTF====
 
The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) of the Arrhenius-exponential is given by,

 
::<math>\begin{align}
& \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t,V)dt=\mathop{}_{0}^{\infty }t\cdot \frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}dt \\
& = & C{{e}^{\tfrac{B}{V}}}
\end{align}</math>

 
====Median====
 
The median, <math>breve{T},</math>
of the Arrhenius-exponential model is given by:

 
::<math>\breve{T}=0.693\cdot C{{e}^{\tfrac{B}{V}}}</math>

 

====Mode====
 
The mode, <math>\tilde{T},</math>
of the Arrhenius-exponential model is given by:

 
::<math>\tilde{T}=0</math>

 

====Standard Deviation====
The standard deviation, <math>{{\sigma }_{T}}</math> , of the Arrhenius-exponential model is given by:

 
::<math>{{\sigma }_{T}}=C{{e}^{\tfrac{B}{V}}}</math>

====Arrhenius-Exponential Reliability Function====
 

The Arrhenius-exponential reliability function is given by:

 
::<math>R(T,V)={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}</math>

 
This function is the complement of the Arrhenius-exponential cumulative distribution function or:

 
::<math>R(T,V)=1-Q(T,V)=1-\mathop{}_{0}^{T}f(T,V)dT</math>

 
and:

 
::<math>R(T,V)=1-\mathop{}_{0}^{T}\frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}dT={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}</math>

 

====Conditional Reliability====
The Arrhenius-exponential conditional reliability function is given by,

::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}</math>

====Reliable Life====
 
For the Arrhenius-exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},</math> is given by:

 
::<math>R({{t}_{R}},V)={{e}^{-\tfrac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}}}</math>

 
::<math>\ln [R({{t}_{R}},V)]=-\frac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}</math>

 
or:

 
::<math>{{t}_{R}}=-C{{e}^{\tfrac{B}{V}}}\ln [R({{t}_{R}},V)]</math>