# Survival function

The survival function, also known as a survivor function or reliability function, is a property of any random variable that maps a set of events, usually associated with mortality or failure of some system, onto time. It captures the probability that the system will survive beyond a specified time.

The term reliability function is common in engineering while the term survival function is used in a broader range of applications, including human mortality. Another name for the survival function is the complementary cumulative distribution function.

## Definition

Let T be a continuous random variable with cumulative distribution function F(t) on the interval [0,∞). Its survival function or reliability function is:

${\displaystyle S(t)=P(\{T>t\})=\int _{t}^{\infty }f(u)\,du=1-F(t).}$

## Properties

Every survival function S(t) is monotonically decreasing, i.e. ${\displaystyle S(u)\leq S(t)}$ for all ${\displaystyle u>t}$.

The time, t = 0, represents some origin, typically the beginning of a study or the start of operation of some system. S(0) is commonly unity but can be less to represent the probability that the system fails immediately upon operation.

Since the CDF is a right-continuous function, the survival function ${\displaystyle S(t)=1-F(t)}$ is also right-continuous.