Editing Survival analysis (section)

===Survival function===
{{main|Survival function}}

The object of primary interest is the '''survival function''', conventionally denoted ''S'', which is defined as

<math display="block">S(t) = \Pr(T > t)</math>
where ''t'' is some time, ''T'' is a [[random variable]] denoting the time of death, and "Pr" stands for [[probability]]. That is, the survival function is the probability that the time of death is later than some specified time ''t''.
The survival function is also called the ''survivor function'' or ''survivorship function'' in problems of biological survival, and the ''reliability function'' in mechanical survival problems. In the latter case, the reliability function is denoted ''R''(''t'').

Usually one assumes ''S''(0) = 1, although it could be less than 1{{nbsp}}if there is the possibility of immediate death or failure.

The survival function must be non-increasing: ''S''(''u'') ≤ ''S''(''t'') if ''u'' ≥ ''t''. This property follows directly because ''T''>''u'' implies ''T''>''t''. This reflects the notion that survival to a later age is possible only if all younger ages are attained. Given this property, the lifetime distribution function and event density (''F'' and ''f'' below) are well-defined.

The survival function is usually assumed to approach zero as age increases without bound (i.e., ''S''(''t'') → 0 as ''t'' → ∞), although the limit could be greater than zero if eternal life is possible. For instance, we could apply survival analysis to a mixture of stable and unstable [[Carbon#Isotopes|carbon isotopes]]; unstable isotopes would decay sooner or later, but the stable isotopes would last indefinitely.