# 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.

## Definition

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

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

## Properties

Every survival function R(t) is monotone decreasing, i.e. ${\displaystyle R(u) for ${\displaystyle u>t}$

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