Template:Weibull mean: Difference between revisions

Latest revision as of 01:39, 13 August 2012

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-=== The Mean or MTTF ===
+#REDIRECT [[Weibull Distribution Functions]]
-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>