# 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:

## Properties

Every survival function *R*(*t*) is monotone decreasing, i.e. for

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.

## See also