|
|
| Line 1: |
Line 1: |
| ===Generalized Gamma Reliability Function===
| | #REDIRECT [[The_Generalized_Gamma_Distribution]] |
| The reliability function for the generalized gamma distribution is given by:
| |
| | |
| ::<math>R(t)=\left\{ \begin{array}{*{35}{l}}
| |
| 1-{{\Gamma }_{I}}\left( \tfrac{{{e}^{\lambda \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}}}{{{\lambda }^{2}}};\tfrac{1}{{{\lambda }^{2}}} \right)\text{ if }\lambda >0 \\
| |
| 1-\Phi \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)\text{ if }\lambda =0 \\
| |
| {{\Gamma }_{I}}\left( \tfrac{{{e}^{\lambda \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}}}{{{\lambda }^{2}}};\tfrac{1}{{{\lambda }^{2}}} \right)\text{ if }\lambda <0 \\
| |
| \end{array} \right.</math>
| |
| | |
| where:
| |
| | |
| ::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z}{{e}^{-\tfrac{{{x}^{2}}}{2}}}dx</math>
| |
| | |
| and <math>{{\Gamma }_{I}}(k;x)</math> is the incomplete gamma function of <math>k</math>
| |
| | |
| and <math>x</math> , which is given by:
| |
| | |
| ::<math>{{\Gamma }_{I}}(k;x)=\frac{1}{\Gamma (k)}\int_{0}^{x}{{s}^{k-1}}{{e}^{-s}}ds</math>
| |
| | |
| where <math>\Gamma (x)</math> is the gamma function of <math>x</math> .
| |
| Note that in Weibull++ the probability plot of the generalized gamma is created on lognormal probability paper. This means that the fitted line will not be straight unless <math>\lambda =0.</math>
| |