Reliability and Survival Analysis

Abstract

We now consider the IG (µ,λ) law from the point of view of modelling for reliability and survival analysis. The reliability of a system at time t is defined as the probability of the system lasting at least until a time t. Thus if X represents failure time,then symbolically R(t) the reliability is given by P(X ≥ t). Since the random variable X has a distribution indexed by a parameter θ it is more convenient to write R(t; θ) for the reliability function. For the IG distribution we will write from now on R(t; µ,λ) to denote this function. Since

$$ \begin{array}{*{20}{c}} {F(t) = \Phi \left[ {\sqrt {\frac{\lambda }{t}\left( {\frac{{\mathop{\rm t}\nolimits} }{\mu } - 1} \right)} } \right] + \exp \left( {\frac{{2\lambda }}{\mu }} \right)\Phi \left[ {\sqrt {\frac{\lambda }{t}} \left( {\sqrt {\frac{t}{\mu }} - 1} \right)} \right],}\\ {R(t;\mu ,\lambda ) = \Phi \left[ {\sqrt {\frac{\lambda }{t}} \left( {1 - \sqrt {\frac{t}{\mu }} } \right)} \right] - \exp \left( {\frac{{2\lambda }}{\mu }} \right)\;}\\ {\Phi \left[ { - \sqrt {\frac{\lambda }{t}} \left( {1 + \sqrt {\frac{t}{\mu }} } \right)} \right]\cdot} \end{array} $$

(5.1)