# Survival Function

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## Definition: Survival Function

Survival Function describes the probability that a variable T takes which is greater than number t. In other words, survival function is defined as the probability that a subject survives beyond time t without experiencing any event of interest. It is given by S(t) which is represented as below

S(t) =  P(T>t)=  tʃαf(u)du = 1- F(t)

Generally, S(0) = 1 but maybe less than 1 in case of immediate death or failure. Survival function is non-increasing and is depicted as S(u) ≤ S(t) provided u ≥ t (Since, T>u implies T>t). It tells that survival to a later age is only possible if all younger ages are attained. Using this property, Lifetime Distribution function and event density are well defined. As “t” tends to infinity, S(t) tends to zero.

Survival function is also called as Reliability function in regard with mechanical survival problems and sometimes it is also called as Complementary cumulative distribution function.

Example

Ram was waiting for his friend Ramesh in the park. Ramesh called and informed Ram that he will be late by ‘X’ minutes after 5pm but will reach before 8pm. Suppose, Ramesh arrives in park at ‘X’ minutes past 5pm, then ‘X’ is uniformly distributed over the interval (0,300).

Then the survival function is given by

S(t)=P(X>t) = 1 if t<0

= 300-t if 0 ≤ t < 300

= 0 if t ≥ 300

Hence, this concludes the definition of Survival Function along with its overview.

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