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Richard House: Redirected page to Distributions Used in Accelerated Testing#The Weibull Distribution

2012-08-16T01:44:51Z

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	#REDIRECT [[Distributions_Used_in_Accelerated_Testing]]		#REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Weibull_Distribution]]

2012-08-09T10:18:05Z

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			#REDIRECT [[Distributions_Used_in_Accelerated_Testing]]
	~~===Statistical Properties Summary===~~
	~~{{aw mean}}~~

	~~{{aw median}}~~

	~~{{aw mode}}~~

	~~{{aw standard deviation}}~~

	~~{{aw cdf and rf}}~~

	~~{{aw crf}}~~

	~~{{aw reliable life}}~~

	~~{{aw frf}}~~

2012-01-12T18:44:25Z

The Failure Rate Function

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	~~====The Failure Rate Function====~~		{{aw frf}}
	~~The 2-parameter Weibull failure rate function,<math>\lambda (T)</math> ,~~
	~~is given by:~~

	~~::<math>\lambda \left( T \right)=\frac~~{~~f\left( T \right)}{R\left( T \right)}=\frac{\beta }{\eta }{{\left( \frac{T}{\eta } \right)}^~~{~~\beta -1~~}}~~</math>~~

2012-01-12T18:44:06Z

The Reliable Life

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	{{aw crf}}		{{aw crf}}

	~~====The Reliable Life====~~		{{aw reliable life}}
	~~For the 2-parameter Weibull distribution, the reliable life, <math>~~{{~~T}_{R}}</math> , of a unit for a specified reliability, starting the mission at age zero, is given by:~~

	~~::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ R\left( {{T}_{R}} \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>~~

	~~This is the~~ life ~~for which the unit will function successfully with a reliability of <math>R({{T~~}~~_{R}})</math> . If <math>R({{T~~}~~_{R}})=0.50</math> then <math>{{T}_{R}}=\breve{T}</math>,~~
	~~the median life, or the life by which half of the units will survive.~~
	~~<br>~~

	====The Failure Rate Function====		====The Failure Rate Function====

2012-01-12T18:43:45Z

The Conditional Reliability Function

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	{{aw cdf and rf}}		{{aw cdf and rf}}

	~~====The Conditional Reliability Function====~~		{{aw crf}}
	~~The Weibull conditional reliability function is given by:~~

	~~::<math>R(T,t)=\frac~~{~~R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}</math>~~

	~~:or:~~

	~~::<math>R(T,t)={{e}^{-\left[ {{\left( \tfrac{T+t}~~{~~\eta } \right)}^{\beta }}-{{\left( \tfrac{T}{\eta } \right)}^{\beta }} \right]~~}}~~</math>~~

	Eqn. (e137) gives the reliability for a new mission of <math>t</math> duration, having already accumulated <math>T</math> hours of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. (It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit(s) already accumulated <math>T</math> hours of operation successfully.)

	====The Reliable Life====		====The Reliable Life====

2012-01-12T18:43:26Z

The cdf and the Reliability Function

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	{{aw standard deviation}}		{{aw standard deviation}}

	~~====The <math>~~cdf~~</math>~~ and ~~the Reliability Function====~~		{{aw cdf and rf}}
	~~The <math>cdf</math> of the 2-parameter Weibull distribution is given by:~~

	~~::<math>F(T)=1-{{e~~}~~^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>~~

	~~The Weibull reliability function is given by:~~

	~~::<math>\begin{align}~~
	~~& R(T)= & 1-F(t) \\~~
	~~& = & {{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}~~
	~~\end{align~~}~~</math>~~

	====The Conditional Reliability Function====		====The Conditional Reliability Function====

2012-01-12T18:43:04Z

The Standard Deviation

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	~~====The Standard Deviation====~~		{{aw standard deviation}}
	~~The~~ standard deviation~~, <math>{{\sigma }_{T}},</math>~~
	~~of the 2-parameter Weibull is given by:~~

	~~::<math>{{\sigma }_{T}}=\eta \cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-\Gamma {{\left( \frac{1}{\beta }+1 \right)}^{2}~~}}~~</math>~~

	====The <math>cdf</math> and the Reliability Function====		====The <math>cdf</math> and the Reliability Function====

2012-01-12T18:42:42Z

The Mode

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	~~====The Mode====~~		{{aw mode}}
	~~The mode, <math>\tilde~~{~~T},</math>~~
	~~of the 2-parameter Weibull is given by:~~
	~~<br>~~
	~~::<math>\tilde~~{~~T}=\eta {{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }~~}}~~</math>~~
	~~<br>~~

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

2012-01-12T18:42:25Z

The Median

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	{{aw mean}}		{{aw mean}}

	~~====The Median====~~		{{aw median}}
	~~The median, <math>\breve~~{~~T},</math>~~
	~~of the 2-parameter Weibull is given by:~~
	~~<br>~~
	~~::<math>\breve~~{~~T}=\eta {{\left( \ln 2 \right)}^{\tfrac{1}{\beta }~~}}~~</math>~~
	~~<br>~~

	====The Mode====		====The Mode====

2012-01-12T18:41:11Z

The Mean or MTTF

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	===Statistical Properties Summary===		===Statistical Properties Summary===
	~~====The Mean or MTTF====~~		{{aw mean}}
	~~The mean, <math>\overline{T},</math> of the 2-parameter Weibull <math>pdf</math> is given by:~~
	~~<br>~~
	~~::<math>\overline~~{~~T}=\eta \cdot \Gamma \left( \frac~~{1}~~{\beta }+1 \right)</math>~~
	~~<br>~~
	~~where <math>\Gamma \left( \tfrac{1}{\beta }+1 \right)</math> is the gamma function evaluated at the value of <math>\left( \tfrac{1}{\beta~~ }~~+1 \right)</math> .~~
	~~<br>~~
	====The Median====		====The Median====
	The median, <math>\breve{T},</math>		The median, <math>\breve{T},</math>