Failure Time Distribution

Let T be a nonnegative random variable that represents the failure time of an individual from a homogeneous superpopulation. The survival distribution function (also known as the survivor function) of T is written as

\[  S(t)=\mr {Pr}(T \geq t)  \]

A mathematically equivalent way of specifying the distribution of T is through its hazard function. The hazard function $\lambda (t)$ specifies the instantaneous failure rate at t. If T is a continuous random variable, $\lambda (t)$ is expressed as

\[  \lambda (t)= \lim _{\Delta t \rightarrow 0^{+} } \frac{ \mr {Pr}(t \leq T < t + \Delta t\  |\  T \geq t) }{ \Delta t } = \frac{f(t)}{S(t)}  \]

where $f(t)$ is the probability density function of T.