Failure Time Distribution
Let T be a nonnegative random variable representing the failure
time of an individual from a homogeneous population. The survival
distribution function (also known as the survivor function)
of T is written as
![S(t)={\rm Pr}(T \geq t)](images/phreq41.gif)
A mathematically equivalent way of specifying the distribution
of T is through its hazard function. The hazard function h(t)
specifies
the instantaneous failure rate at t. If T is a continuous random variable,
h(t) is expressed as
![h(t)= \lim_{\Delta t arrow 0^{+} }
\frac{ {\rm Pr}(t \leq T \lt t + \Delta t|T \geq t) }{ \Delta t }
= \frac{f(t)}{S(t)}](images/phreq42.gif)
where f(t) is the probability density function of T. If T is discrete
with masses at x1 < x2 < ... , then h(t) is given by
![h(t)=\sum_{j} h_{j} \delta (t- x_{j})](images/phreq43.gif)
where
![\delta(u)=\{ 0 & {\rm if}u \lt 0 \ 1 & {\rm otherwise}
.](images/phreq44.gif)
![h_{j}={\rm Pr}(T=x_{j}|T \geq x_{j})=\frac{ {\rm Pr}(T=x_{j}) }{S(x_{j}) }](images/phreq45.gif)
for j = 1, 2, ...
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.