Survival analysis deals with that branch of statistics which analyses the time of occurrence of certain events – such as failure in a machine, death of a person etc. The hazard function is the ratio of density function and survival function.
where T is the survival model of a system being studied
T is taken as a continuous random variable
Discrete time hazard function is the conditional probability that individual i will experience the target event in Time period (Ti =j) where the individual has not experienced the event so far.
Since it is a probability in discrete time , the value of hazard function lies between 0 and 1.
Suppose a doctor A is informed that his patient will arrive between 9am and 2 pm for the surgery. Suppose patient B arrives X minutes past 9, then X is distributed between (0,300)
The probability density function is given
P(X) = 0
Otherwise 1 given 0<t<300
The survival function is given by
F-t=P(X) =1 if t<0
300-t if 0<t<=300
1 if t>300
So the hazard function can be defined as ratio of density and survival function for any time t.
At 9 am, hazard function= 0.003