Weibull Distribution Functions - Revision history

Lisa Hacker at 21:40, 18 September 2023

2023-09-18T21:40:52Z

← Older revision		Revision as of 21:40, 18 September 2023
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	<noinclude>{{Navigation box}}		<noinclude>{{Navigation box}}
	''This article also appears in the [~~[The_Weibull_Distribution\|~~Life ~~Data Analysis Reference]~~] and [~~[Distributions_Used_in_Accelerated_Testing\|~~Accelerated ~~Life Testing Data Analysis Reference]~~] ~~books~~.'' </noinclude>		''This article also appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference] and [https://help.reliasoft.com/reference/accelerated_life_testing_data_analysis Accelerated life testing reference].'' </noinclude>
	=== The Mean or MTTF ===		=== The Mean or MTTF ===
	The mean, <math> \overline{T} \,\!</math>, (also called ''MTTF'') of the Weibull ''pdf'' is given by:		The mean, <math> \overline{T} \,\!</math>, (also called ''MTTF'') of the Weibull ''pdf'' is given by:

Richard House: /* The Weibull Reliable Life */

2012-09-24T22:51:30Z

The Weibull Reliable Life

← Older revision		Revision as of 22:51, 24 September 2012
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	=== The Weibull Reliable Life ===		=== The 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:		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>		::<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 ''R''. If ''R''=0.50, then <math> T_{R}=\breve{T} \,\!</math>, the median life, or the life by which half of the units will survive.		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 ===

Richard House at 18:18, 24 September 2012

2012-09-24T18:18:12Z

Melinda Caroline: /* The Weibull Conditional Reliability Function */

2012-08-14T05:55:46Z

The Weibull Conditional Reliability Function

← Older revision		Revision as of 05:55, 14 August 2012
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	::<math> R(t\|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } </math>		::<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.		These give 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 ===

Lisa Hacker: /* The Weibull Reliable Life */

2012-08-08T07:53:15Z

The Weibull Reliable Life

← Older revision		Revision as of 07:53, 8 August 2012
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	=== The Weibull Reliable Life ===		=== The Weibull Reliable Life ===
	The reliable life, <math> T_{R} \,\!</math>, of a unit for a specified reliability, ''R'', starting the mission at age zero, is given by:		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>		::<math> T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} </math>

Lisa Hacker: /* The Standard Deviation */

2012-08-08T07:52:21Z

The Standard Deviation

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	=== The Standard Deviation ===		=== The Standard Deviation ===
	The standard deviation, <math> \sigma _{T}</math>, is given by:		The standard deviation, <math> \sigma _{T}\,\!</math>, 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>		::<math> \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} </math>

Lisa Hacker: /* The Standard Deviation */

2012-08-08T07:51:39Z

The Standard Deviation

← Older revision		Revision as of 07:51, 8 August 2012
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	=== The Standard Deviation ===		=== The Standard Deviation ===
	The standard deviation, <~~span class="texhtml"~~>~~σ<sub>''~~T~~''</sub>~~</~~span~~>, is given by:		The standard deviation, <math> \sigma _{T}</math>, 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>		::<math> \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} </math>

Lisa Hacker: /* The Weibull Reliability Function */

2012-08-07T03:33:03Z

The Weibull Reliability Function

← Older revision		Revision as of 03:33, 7 August 2012
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	The equation for the 3-parameter Weibull cumulative density function, ''cdf'', is given by:		The equation for the 3-parameter Weibull cumulative density function, ''cdf'', is given by:

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

	This is also referred to as ''unreliability'' and designated as <math> Q(t) \,\!</math> by some authors.		This is also referred to as ''unreliability'' and designated as <math> Q(t) \,\!</math> by some authors.

Lisa Hacker: /* The Mean or MTTF */

2012-08-07T03:32:41Z

The Mean or MTTF

← Older revision		Revision as of 03:32, 7 August 2012
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	Note that some practitioners erroneously assume that <math> \eta \,\!</math> is equal to the MTTF, <math> \overline{T}\,\!</math>.		Note that some practitioners erroneously assume that <math> \eta \,\!</math> is equal to the MTTF, <math> \overline{T}\,\!</math>.
	This is only true for the case of:		This is only true for the case of:
	<math> \beta=1 \,\!</math> or		<math> \beta=1 \,\!</math> or:

	::<math>		::<math>

Lisa Hacker: /* The Mean or MTTF */

2012-08-07T03:32:24Z

The Mean or MTTF

← Older revision		Revision as of 03:32, 7 August 2012
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	is the gamma function evaluated at the value of:		is the gamma function evaluated at the value of:

	::<math> \left( { \frac{1}{\beta }}+1\right) \,\!</math>.		::<math> \left( { \frac{1}{\beta }}+1\right) \,\!</math>

	The gamma function is defined as:		The gamma function is defined as:
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	Note that some practitioners erroneously assume that <math> \eta \,\!</math> is equal to the MTTF, <math> \overline{T}\,\!</math>.		Note that some practitioners erroneously assume that <math> \eta \,\!</math> is equal to the MTTF, <math> \overline{T}\,\!</math>.
	This is only true for the case of		This is only true for the case of:
	<math> \beta=1 \,\!</math> or		<math> \beta=1 \,\!</math> or

← Older revision		Revision as of 18:18, 24 September 2012
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	&= \eta		&= \eta
	\end{align}		\end{align}
	</math>		\,\!</math>

	=== The Median ===		=== The Median ===
	The median, <math> \breve{T}</math>, of the Weibull distribution is given by:		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>		::<math> \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} \,\!</math>

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

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

	=== The Standard Deviation ===		=== The Standard Deviation ===
	The standard deviation, <math> \sigma _{T}\,\!</math>, is given by:		The standard deviation, <math> \sigma _{T}\,\!</math>, 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>		::<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 ===
	The equation for the 3-parameter Weibull cumulative density function, ''cdf'', is given by:		The equation for the 3-parameter Weibull cumulative density function, ''cdf'', is given by:

	::<math> F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} </math>		::<math> F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} \,\!</math>

	This is also referred to as ''unreliability'' and designated as <math> Q(t) \,\!</math> by some authors.		This is also referred to as ''unreliability'' and designated 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 3-parameter Weibull distribution is then given by:		Recalling that the reliability function of a distribution is simply one minus the ''cdf'', the reliability function for the 3-parameter Weibull distribution is then given by:
	::<math> R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} </math>		::<math> R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\!</math>

	=== The Weibull Conditional Reliability Function ===		=== The Weibull Conditional Reliability Function ===
	The 3-parameter Weibull conditional reliability function is given by:		The 3-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>		::<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:		or:

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

	These give 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.		These give 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 ===
	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:		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>		::<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 ''R''. If ''R''=0.50, then <math> T_{R}=\breve{T} </math>, the median life, or the life by which half of the units will survive.		This is the life for which the unit/item will be functioning successfully with a reliability of ''R''. If ''R''=0.50, 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 ===
	The Weibull failure rate function, <math> \lambda(t) \,\!</math>, is given by:		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>		::<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>