# Definition:Survival Function

## Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f: X \to \overline \R$ be a $\Sigma$-measurable function.

The **survival function of $f$** is the mapping $F_f: \R \to \overline \R$ defined by:

- $\forall t \in \R: \map {F_f} t := \map \mu {\set {\size f \ge t} }$

where $\set {\size f \ge t}$ denotes the set $\set {x \in X: \size {\map f x} \ge t}$.

## Also known as

Some sources refer to this as a **distribution function**, but it can then become confused with the concept of a distribution function in physics.

Also, this term could be confused with the cumulative distribution function, which is closely related.

Further alternatives include **survivor function**, **reliability function** and **complementary cumulative distribution function**.

All terms (including **survival function** itself) have their origin in probability theory.

## Also denoted as

The survival function may also be denoted by $d_f$ (for the "distribution function" of $f$) or $\map F f$.

For the sake of consistency, $F_f$ is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

- Results about
**survival functions**can be found**here**.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 13$ - 2014: Loukas Grafakos:
*Classical Fourier Analysis*(3rd ed.) ... (previous) ... (next): $1.1.1$: The Distribution Function