Weibull Distribution Functions: Difference between revisions
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<noinclude>{{Navigation box}} | <noinclude>{{Navigation box}} | ||
''This article also appears in the [ | ''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: | ||
| Line 11: | Line 11: | ||
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: | ||
| Line 22: | Line 22: | ||
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> | ||
| Line 33: | Line 33: | ||
&= \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, < | 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 | 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, | 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 | 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 === | ||
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> | ||
Latest revision as of 21:40, 18 September 2023
This article also appears in the Life data analysis reference and Accelerated life testing reference.
The Mean or MTTF
The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF) of the Weibull pdf is given by:
- [math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]
where
- [math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]
is the gamma function evaluated at the value of:
- [math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math]
The gamma function is defined as:
- [math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]
For the 2-parameter case, this can be reduced to:
- [math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]
Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of: [math]\displaystyle{ \beta=1 \,\! }[/math] or:
- [math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} \,\! }[/math]
The Median
The median, [math]\displaystyle{ \breve{T}\,\! }[/math], of the Weibull distribution is given by:
- [math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} \,\! }[/math]
The Mode
The mode, [math]\displaystyle{ \tilde{T} \,\! }[/math], is given by:
- [math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} \,\! }[/math]
The Standard Deviation
The standard deviation, [math]\displaystyle{ \sigma _{T}\,\! }[/math], is given by:
- [math]\displaystyle{ \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 equation for the 3-parameter Weibull cumulative density function, cdf, is given by:
- [math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} \,\! }[/math]
This is also referred to as unreliability and designated as [math]\displaystyle{ 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:
- [math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]
The Weibull Conditional Reliability Function
The 3-parameter Weibull conditional reliability function is given by:
- [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ 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 reliable life, [math]\displaystyle{ T_{R}\,\! }[/math], of a unit for a specified reliability, [math]\displaystyle{ R\,\! }[/math], starting the mission at age zero, is given by:
- [math]\displaystyle{ 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]\displaystyle{ R\,\! }[/math]. If [math]\displaystyle{ R = 0.50\,\! }[/math], then [math]\displaystyle{ 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, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:
- [math]\displaystyle{ \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]