# 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.