Template:Weibull mean

The Mean or MTTF
The mean, $$ \overline{T} \,\!$$, (also called MTTF of the Weibull pdf is given by:


 * $$ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\!$$

where
 * $$ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\!$$

is the gamma function evaluated at the value of


 * $$ \left( { \frac{1}{\beta }}+1\right) \,\!$$.

The gamma function is defined as:


 * $$ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\!$$

For the two-parameter case, this can be reduced to:


 * $$ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\!$$

Note that some practitioners erroneously assume that $$ \eta \,\!$$ is equal to the MTTF, $$ \overline{T}\,\!$$. This is only true for the case of $$ \beta=1 \,\!$$ or



\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} $$