Template:Weibull mean: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
(Created page with '=== 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 \…')
 
Line 5: Line 5:
::<math> \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right)  \,\!</math>
::<math> \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right)  \,\!</math>


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



Revision as of 21:29, 9 February 2012

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 two-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]