Template:Generalized gamma reliability function: Difference between revisions

Latest revision as of 09:36, 9 August 2012

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-===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}}
--{{\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  \\
--\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>