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== Weibull Statistical Properties ==
#REDIRECT [[Weibull Distribution Functions]]
 
=== The Mean or MTTF ===
 
The mean, <math> \overline{T} \,\!</math>, (also called ''MTTF'' of the Weibull ''pdf'' is given by:
 
::<math> \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right)  \,\!</math>
 
:where
::<math> \Gamma \left( {\frac{1}{\beta }}+1\right) \,\!</math>
 
is the gamma function evaluated at the value of
 
::<math> \left( { \frac{1}{\beta }}+1\right) \,\!</math>.
 
The gamma function is defined as:
 
::<math> \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\!</math>
 
For the two-parameter case, this can be reduced to:
 
::<math> \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\!</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
<math> \beta=1 \,\!</math> or
 
::<math>
\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> \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, <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, <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 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 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 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, <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>

Latest revision as of 07:52, 10 August 2012

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