Template:Eyring-ex stat prop sum - Revision history

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

2012-08-16T22:59:18Z

Redirected page to Eyring Relationship#Eyring-Exponential

← Older revision		Revision as of 22:59, 16 August 2012
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	~~===~~Eyring-Exponential ~~Statistical Properties Summary===~~		#REDIRECT [[Eyring_Relationship#Eyring-Exponential]]
	~~{{eyring-ex mean}}~~

	~~{{eyring-ex median}}~~

	~~{{eyring-ex mode}}~~

	~~{{eyring-ex sd}}~~

	~~{{eyring-ex rf}}~~

	~~{{eyring-ex cr}}~~

	~~{{eyring-ex rl}}~~

Nicolette Young: /* Reliable Life */

2012-01-12T22:37:59Z

Reliable Life

← Older revision		Revision as of 22:37, 12 January 2012
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	{{eyring-ex cr}}		{{eyring-ex cr}}

	~~====Reliable Life====~~		{{eyring-ex rl}}
	~~<br>~~
	~~For the Eyring-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}^~~{-~~{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>~~
	~~<br>~~
	~~::<math>\ln [R({{t}_{R}},V)]=-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}</math>~~
	~~<br>~~
	~~:or:~~
	~~<br>~~
	~~::<math>{{t}_{R}}=-\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\ln [R({{t}_{R~~}}~~,V)]</math>~~

Nicolette Young: /* Conditional Reliability */

2012-01-12T22:37:36Z

Conditional Reliability

← Older revision		Revision as of 22:37, 12 January 2012
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	{{eyring-ex rf}}		{{eyring-ex rf}}

	~~====Conditional Reliability====~~		{{eyring-ex cr}}
	~~<br>~~
	~~The conditional reliability function for the Eyring-exponential model is given by:~~
	~~<br>~~
	~~::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-t\cdot V\cdot~~ {{~~e}^{\left( A~~-~~\tfrac{B}{V} \right)}}~~}}~~</math>~~
	~~<br>~~
	====Reliable Life====		====Reliable Life====
	<br>		<br>

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

2012-01-12T22:37:16Z

Eyring-Exponential Reliability Function

@@ Line 8: / Line 8: @@
 {{eyring-ex sd}}
-====Eyring-Exponential Reliability Function====
+{{eyring-ex rf}}
 ====Conditional Reliability====

Nicolette Young: /* Standard Deviation */

2012-01-12T22:36:56Z

Standard Deviation

← Older revision		Revision as of 22:36, 12 January 2012
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	{{eyring-ex mode}}		{{eyring-ex mode}}

	~~====Standard Deviation====~~		{{eyring-ex sd}}
	~~<br>~~
	~~The standard deviation, <math>{{\sigma }_{T}}</math>, for the Eyring-exponential model is given by:~~
	~~<br>~~
	~~::<math>{{\sigma }_{T}}=\frac{1}{V}{~~{~~e}^~~{-~~\left( A-\tfrac{B}{V} \right)~~}}~~</math>~~
	~~<br>~~
	====Eyring-Exponential Reliability Function====		====Eyring-Exponential Reliability Function====
	<br>		<br>

Nicolette Young: /* Mode */

2012-01-12T22:36:37Z

Mode

← Older revision		Revision as of 22:36, 12 January 2012
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	{{eyring-ex median}}		{{eyring-ex median}}

	~~====Mode====~~		{{eyring-ex mode}}
	~~<br>~~
	~~The~~ mode~~, <math>\tilde{T~~}~~,</math>~~
	~~for the Eyring-exponential model is <math>\tilde{T~~}~~=0.</math>~~
	~~<br>~~
	====Standard Deviation====		====Standard Deviation====
	<br>		<br>

Nicolette Young: /* Median */

2012-01-12T22:36:14Z

Median

← Older revision		Revision as of 22:36, 12 January 2012
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	{{eyring-ex mean}}		{{eyring-ex mean}}

	~~====Median====~~		{{eyring-ex median}}
	~~<br>~~

	~~The median, <math>\breve{T},</math>~~
	~~for the Eyring-exponential model is given by:~~

	~~<br>~~

	~~::<math>\breve{T}=0.693\frac{1}{V}~~{{~~e}^{~~-~~\left( A-\tfrac{B}{V} \right)~~}}~~</math>~~

	~~<br>~~

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

Nicolette Young: /* Mean or MTTF */

2012-01-12T22:34:26Z

Mean or MTTF

← Older revision		Revision as of 22:34, 12 January 2012
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	===Eyring-Exponential Statistical Properties Summary===		===Eyring-Exponential Statistical Properties Summary===
	~~====Mean or MTTF====~~		{{eyring-ex mean}}
	~~<br>~~
	~~The~~ mean~~, <math>\overline{T~~}~~,</math> or Mean Time To Failure (MTTF) for the Eyring-exponential is given by:~~

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

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

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

2012-01-12T22:33:31Z

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

New page

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

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

====Median====
 

The median, <math>\breve{T},</math>
for the Eyring-exponential model is given by:

 

::<math>\breve{T}=0.693\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>

 

====Mode====
 
The mode, <math>\tilde{T},</math>
for the Eyring-exponential model is <math>\tilde{T}=0.</math>
 
====Standard Deviation====
 
The standard deviation, <math>{{\sigma }_{T}}</math>, for the Eyring-exponential model is given by:
 
::<math>{{\sigma }_{T}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
 
====Eyring-Exponential Reliability Function====
 
The Eyring-exponential reliability function is given by:

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

 
This function is the complement of the Eyring-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}V{{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}dT={{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>

====Conditional Reliability====
 
The conditional reliability function for the Eyring-exponential model 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}^{-t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
 
====Reliable Life====
 
For the Eyring-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}^{-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
 
::<math>\ln [R({{t}_{R}},V)]=-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}</math>
 
:or:
 
::<math>{{t}_{R}}=-\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\ln [R({{t}_{R}},V)]</math>