Template:Weibull Statistical Properties - Revision history

Lisa Hacker: Redirected page to Weibull Distribution Functions

2012-08-10T07:52:02Z

← Older revision		Revision as of 07:52, 10 August 2012
Line 1:		Line 1:
	== Weibull ~~Statistical Properties ==~~		#REDIRECT [[Weibull Distribution Functions]]

	~~{{weibull mean}}~~

	~~{{weibull median}}~~

	~~{{weibull mode}}~~

	~~{{weibull standard deviation}}~~

	~~{{weibull reliability function}}~~

	~~{{weibull conditional reliability function}}~~

	~~{{weibull reliable life}}~~

	~~{{weibull failure rate function}}~~

2012-01-04T16:35:36Z

The Weibull Failure Rate Function

← Older revision		Revision as of 16:35, 4 January 2012
Line 15:		Line 15:
	{{weibull reliable life}}		{{weibull reliable life}}

	~~=== The Weibull Failure Rate Function ===~~		{{weibull failure rate function}}

	~~The 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-\gamma~~ }~~{\eta }\right) ^{\beta -1~~} ~~</math>~~

2012-01-04T16:35:03Z

The Weibull Reliable Life

← Older revision		Revision as of 16:35, 4 January 2012
Line 13:		Line 13:
	{{weibull conditional reliability function}}		{{weibull conditional reliability function}}

	~~=== The Weibull Reliable Life ===~~		{{weibull reliable life}}

	~~The reliable life, <math> T_~~{~~R} \,\!</math>, of a unit for a specified reliability,<math> R \,\!</math>, starting the mission at age zero, is given by:~~

	~~::<math> T_~~{~~R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} </math>~~

	~~This is the~~ life ~~for which the unit/item will be functioning successfully with a reliability of <math> R \,\!</math>, . If ,<math> 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.~~

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

2012-01-04T16:34:08Z

The Weibull Conditional Reliability Function

← Older revision		Revision as of 16:34, 4 January 2012
Line 11:		Line 11:
	{{weibull reliability function}}		{{weibull reliability function}}

	~~=== The Weibull Conditional Reliability Function ===~~		{{weibull conditional reliability function}}

	~~The three-parameter Weibull~~ conditional reliability function ~~is given by:~~

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

	~~:or:~~

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

	These gives the reliability for a new mission of <math> t \,\!</math> duration, having already accumulated <math> T \,\!</math> time 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 or units already accumulated hours of operation successfully.

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

2012-01-04T16:33:39Z

← Older revision		Revision as of 16:33, 4 January 2012
Line 9:		Line 9:
	{{weibull standard deviation}}		{{weibull standard deviation}}

	{{weibull reliability function		{{weibull reliability function}}

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

2012-01-04T16:33:05Z

The Weibull Reliability Function

← Older revision		Revision as of 16:33, 4 January 2012
Line 9:		Line 9:
	{{weibull standard deviation}}		{{weibull standard deviation}}

	~~=== The Weibull Reliability Function ===~~		{{weibull reliability function

	~~The equation for the three-parameter Weibull cumulative density function, ''cdf'', is given by:~~

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

	~~This is also referred to as ''Unreliability'' and deignated as <math> Q(T) \,\!</math> by some authors.~~


	~~Recalling that the reliability function of a distribution is simply one minus the cdf, the~~ reliability function ~~for the three-parameter Weibull distribution is then given by:~~
	~~::<math> R(T)=e^{-\left( { \frac{T-\gamma }{\eta }}\right) ^{\beta }} </math>~~

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

2012-01-04T16:32:40Z

The Standard Deviation

← Older revision		Revision as of 16:32, 4 January 2012
Line 7:		Line 7:
	{{weibull mode}}		{{weibull mode}}

	~~=== The Standard Deviation ===~~		{{weibull standard deviation}}

	~~The~~ standard deviation~~, <span class="texhtml">σ<sub>''T''</sub>,</span> 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 Weibull Reliability Function ===		=== The Weibull Reliability Function ===

2012-01-04T16:32:14Z

The Mode

← Older revision		Revision as of 16:32, 4 January 2012
Line 5:		Line 5:
	{{weibull median}}		{{weibull median}}

	~~=== The Mode ===~~		{{weibull mode}}

	~~The mode, <math> \tilde~~{~~T}, </math> is given by:~~

	~~::<math> \tilde~~{~~T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta~~ }} ~~</math>~~

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

2012-01-04T16:31:52Z

The Median

← Older revision		Revision as of 16:31, 4 January 2012
Line 3:		Line 3:
	{{weibull mean}}		{{weibull mean}}

	~~=== The Median ===~~		{{weibull median}}

	~~The median, <math> \breve~~{~~T}, </math> of the Weibull distribution is given by:~~

	~~::<math> \breve~~{~~T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta~~ }} ~~</math>~~

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

2012-01-04T16:31:28Z

The Mean or MTTF

@@ Line 1: / Line 1: @@
 == Weibull Statistical Properties ==
-=== The Mean or MTTF ===
+{{weibull mean}}
 === The Median ===