Template:Alta statistical properties summary - Revision history

Richard House: Redirected page to Distributions Used in Accelerated Testing

2012-08-16T01:13:44Z

Redirected page to Distributions Used in Accelerated Testing

← Older revision		Revision as of 01:13, 16 August 2012
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	~~===Statistical Properties Summary===~~		#REDIRECT [[Distributions_Used_in_Accelerated_Testing]]
	~~{{alta exponential mean}}~~

	~~{{alta exponential median}}~~

	~~{{alta exponential mode}}~~

	~~{{alta exponential sd}}~~

	~~{{alta exponential reliability function}}~~

	~~{{alta exponential conditional reliability}}~~

	~~{{alta exponential reliable life}}~~

	~~{{alta exponential frf}}~~

Nicolette Young: /* Failure Rate Function */

2012-01-12T18:12:54Z

Failure Rate Function

← Older revision		Revision as of 18:12, 12 January 2012
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	{{alta exponential reliable life}}		{{alta exponential reliable life}}

	~~====Failure Rate Function====~~		{{alta exponential frf}}
	~~The exponential failure rate function is given by:~~
	~~<br>~~
	~~::<math>\lambda (T)=\frac~~{~~f(T)}~~{~~R(T)}=\frac{\lambda {{e}^{-\lambda (T)}}}{{{e}^{-\lambda (T)~~}}~~}=\lambda =\text{Constant}</math>~~
	~~<br>~~

Nicolette Young: /* Reliable Life */

2012-01-12T18:12:27Z

Reliable Life

← Older revision		Revision as of 18:12, 12 January 2012
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	{{alta exponential conditional reliability}}		{{alta exponential conditional reliability}}

	~~====Reliable Life====~~		{{alta exponential reliable life}}
	~~The~~ reliable life~~, or the mission duration for a desired reliability goal, <math>{{t}_{R~~}}~~</math> , for the 1-parameter exponential distribution is given by:~~
	~~<br>~~

	~~::<math>\begin{align}~~
	~~& R({{t}_{R}})= & {{e}^{-\lambda {{t}_{R}}}} \\~~
	~~& & \\~~
	~~& \ln [R({{t}_{R}})]= & -\lambda {{t}_{R}}~~
	~~\end{align}</math>~~
	~~<br>~~
	~~:or:~~
	~~<br>~~
	~~::<math>{{t}_{R}}=-\frac{\ln [R({{t}_{R}})]}{\lambda }</math>~~

	~~<br>~~
	====Failure Rate Function====		====Failure Rate Function====
	The exponential failure rate function is given by:		The exponential failure rate function is given by:

Nicolette Young: /* Conditional Reliability */

2012-01-12T18:11:39Z

Conditional Reliability

← Older revision		Revision as of 18:11, 12 January 2012
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	{{alta exponential reliability function}}		{{alta exponential reliability function}}

	~~====Conditional Reliability====~~		{{alta exponential conditional reliability}}
	~~<br>~~
	~~The~~ conditional reliability ~~function for the 1-parameter exponential distribution is given by:~~
	~~<br>~~
	~~::<math>R(T,t)=\frac{R(T+t)~~}~~{R(T)~~}~~=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\lambda t}}</math>~~
	~~<br>~~
	which says that the reliability for a mission of <math>t</math> duration undertaken after the component or equipment has already accumulated <math>T</math> hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the ``memoryless property.''
	~~<br>~~
	====Reliable Life====		====Reliable Life====
	The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}</math> , for the 1-parameter exponential distribution is given by:		The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}</math> , for the 1-parameter exponential distribution is given by:

Nicolette Young: /* The Reliability Function */

2012-01-12T18:10:55Z

The Reliability Function

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 {{alta exponential sd}}
-====The Reliability Function====
+{{alta exponential reliability function}}
 ====Conditional Reliability====

Nicolette Young: /* The Standard Deviation */

2012-01-12T18:09:47Z

The Standard Deviation

← Older revision		Revision as of 18:09, 12 January 2012
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	{{alta exponential mode}}		{{alta exponential mode}}

	~~====The Standard Deviation====~~		{{alta exponential sd}}
	~~The standard deviation, <math>~~{{~~\sigma }_{T}}</math> , of the 1-parameter~~ exponential ~~distribution is given by:~~
	~~<br>~~
	~~::<math>{{\sigma }_{T}}=\frac{1~~}~~{\lambda~~ }~~=m</math>~~

	~~<br>~~
	====The Reliability Function====		====The Reliability Function====
	The 1-parameter exponential reliability function is given by:		The 1-parameter exponential reliability function is given by:

Nicolette Young: /* The Mode */

2012-01-12T18:09:26Z

The Mode

← Older revision		Revision as of 18:09, 12 January 2012
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	{{alta exponential median}}		{{alta exponential median}}

	~~====The Mode====~~		{{alta exponential mode}}
	~~The~~ mode~~, <math>\tilde{T~~}~~,</math>~~
	~~of the 1-parameter exponential distribution is given by:~~
	~~<br>~~
	~~::<math>\tilde{T~~}~~=0</math>~~
	~~<br>~~

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

Nicolette Young: /* The Median */

2012-01-12T18:09:07Z

The Median

← Older revision		Revision as of 18:09, 12 January 2012
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	{{alta exponential mean}}		{{alta exponential mean}}

	~~====The Median====~~		{{alta exponential median}}
	~~The median, <math>\breve~~{~~T}</math>,~~
	~~of the 1-parameter~~ exponential ~~distribution is given by:~~
	~~<br>~~
	~~::<math>\breve{T}=\frac{1~~}~~{\lambda~~ }~~0.693</math>~~
	~~<br>~~
	====The Mode====		====The Mode====
	The mode, <math>\tilde{T},</math>		The mode, <math>\tilde{T},</math>

Nicolette Young: /* The Mean or MTTF */

2012-01-12T18:08:42Z

The Mean or MTTF

← Older revision		Revision as of 18:08, 12 January 2012
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	===Statistical Properties Summary===		===Statistical Properties Summary===
	~~====The Mean or MTTF====~~		{{alta exponential mean}}
	~~The mean, <math>\overline~~{~~T},</math> or Mean Time To Failure (MTTF) of the 1-parameter~~ exponential ~~distribution is given by:~~
	~~<br>~~
	~~::<math>\begin{align}~~
	~~& \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t)dt=\mathop{}_{0}^{\infty }t\cdot \lambda \cdot {{e~~}~~^{-\lambda t}}dt \\~~
	~~& = & \frac{1}{\lambda }~~
	~~\end{align~~}~~</math>~~

	====The Median====		====The Median====

Nicolette Young: Created page with '===Statistical Properties Summary=== ====The Mean or MTTF==== The mean, $\overline{T},$ or Mean Time To Failure (MTTF) of the 1-parameter exponential distribution is…'

2012-01-12T18:06:33Z

Created page with '===Statistical Properties Summary=== ====The Mean or MTTF==== The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) of the 1-parameter exponential distribution is…'

New page

===Statistical Properties Summary===
====The Mean or MTTF====
The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) of the 1-parameter exponential distribution is given by:
 
::<math>\begin{align}
& \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t)dt=\mathop{}_{0}^{\infty }t\cdot \lambda \cdot {{e}^{-\lambda t}}dt \\
& = & \frac{1}{\lambda }
\end{align}</math>

====The Median====
The median, <math>\breve{T}</math>,
of the 1-parameter exponential distribution is given by:
 
::<math>\breve{T}=\frac{1}{\lambda }0.693</math>
 
====The Mode====
The mode, <math>\tilde{T},</math>
of the 1-parameter exponential distribution is given by:
 
::<math>\tilde{T}=0</math>
 

====The Standard Deviation====
The standard deviation, <math>{{\sigma }_{T}}</math> , of the 1-parameter exponential distribution is given by:
 
::<math>{{\sigma }_{T}}=\frac{1}{\lambda }=m</math>

 
====The Reliability Function====
The 1-parameter exponential reliability function is given by:
 
::<math>R(T)={{e}^{-\lambda T}}={{e}^{-\tfrac{T}{m}}}</math>
 
This function is the complement of the exponential cumulative distribution function or:
 
::<math>R(T)=1-Q(T)=1-\mathop{}_{0}^{T}f(T)dT</math>
 
:and:
 
::<math>R(T)=1-\mathop{}_{0}^{T}\lambda {{e}^{-\lambda T}}dT={{e}^{-\lambda T}}</math>
 

====Conditional Reliability====
 
The conditional reliability function for the 1-parameter exponential distribution is given by:
 
::<math>R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\lambda t}}</math>
 
which says that the reliability for a mission of <math>t</math> duration undertaken after the component or equipment has already accumulated <math>T</math> hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the ``memoryless property.''
 
====Reliable Life====
The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}</math> , for the 1-parameter exponential distribution is given by:
 

::<math>\begin{align}
& R({{t}_{R}})= & {{e}^{-\lambda {{t}_{R}}}} \\
& & \\
& \ln [R({{t}_{R}})]= & -\lambda {{t}_{R}}
\end{align}</math>
 
:or:
 
::<math>{{t}_{R}}=-\frac{\ln [R({{t}_{R}})]}{\lambda }</math>

 
====Failure Rate Function====
The exponential failure rate function is given by:
 
::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\lambda {{e}^{-\lambda (T)}}}{{{e}^{-\lambda (T)}}}=\lambda =\text{Constant}</math>