Failure Time Distribution

Let 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 is written as


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


where is the probability density function of . If is discrete with masses at , then survivor function is given by


where =0 if < 0 and =1 otherwise. The discrete hazards are given by