Template:Generalized gamma reliability function

Generalized Gamma Reliability Function

The reliability function for the generalized gamma distribution is given by:

[math]\displaystyle{ 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 \gt 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 \lt 0 \\ \end{array} \right. }[/math]

where:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z}{{e}^{-\tfrac{{{x}^{2}}}{2}}}dx }[/math]

and [math]\displaystyle{ {{\Gamma }_{I}}(k;x) }[/math] is the incomplete gamma function of [math]\displaystyle{ k }[/math]

and [math]\displaystyle{ x }[/math] , which is given by:

[math]\displaystyle{ {{\Gamma }_{I}}(k;x)=\frac{1}{\Gamma (k)}\int_{0}^{x}{{s}^{k-1}}{{e}^{-s}}ds }[/math]

where [math]\displaystyle{ \Gamma (x) }[/math] is the gamma function of [math]\displaystyle{ 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]\displaystyle{ \lambda =0. }[/math]