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

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.

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 $\mathsf{Pr} \infty \mathsf{fWiki}$.