Template:Generalized gamma reliability function

Generalized Gamma Reliability Function
The reliability function for the generalized gamma distribution is given by:


 * $$R(t)=\left\{ \begin{array}{*{35}{l}}

1-{{\Gamma }_{I}}\left( \tfrac;\tfrac{1} \right)\text{ if }\lambda >0 \\ 1-\Phi \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)\text{              if }\lambda =0  \\ {{\Gamma }_{I}}\left( \tfrac;\tfrac{1} \right)\text{      if }\lambda <0  \\ \end{array} \right.$$

where:


 * $$\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z}{{e}^{-\tfrac{2}}}dx$$

and $${{\Gamma }_{I}}(k;x)$$  is the incomplete gamma function of  $$k$$

and $$x$$, which is given by:


 * $${{\Gamma }_{I}}(k;x)=\frac{1}{\Gamma (k)}\int_{0}^{x}{{s}^{k-1}}{{e}^{-s}}ds$$

where $$\Gamma (x)$$  is the gamma function of  $$x$$. 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 $$\lambda =0.$$