= Survival function
{title2=$S(t)$}
{wiki}
= Survivor function
{synonym}
For a nonnegative <survival time> $T$, the <survival function> is
$$
S(t)=\mathbb P(T>t)=1-F(t),\qquad F(t)=\mathbb P(T\leq t),
$$
where $F$ is the <cumulative distribution function>. Thus $S(t)$ is the <probability> that the chosen occurrence is still in the future at time $t$. We consistently use the strict inequality $T>t$: a <probability mass> at $t$ has already left the <survival function> at $t$ itself. With this convention $S$ is a <right-continuous function> and a <nonincreasing function> taking values in $[0,1]$. If $\mathbb P(T=0)=0$, then $S(0)=1$; if $T$ is finite with <probability> one, then $S(t)\to0$ as $t\to\infty$.
For $S(a)>0$ and $b>a$, <conditional probability> gives
$$
\mathbb P(T>b\mid T>a)=\frac{S(b)}{S(a)},\qquad
\mathbb P(a<T\leq b\mid T>a)=1-\frac{S(b)}{S(a)}.
$$
For example, $S(2)=0.8$ and $S(5)=0.6$ mean that $0.6/0.8=0.75$ of those still event-free at year two remain event-free through year five. Their intervening event <probability> is $0.25$, whereas the original population's <probability> of an occurrence in $(2,5]$ is $0.8-0.6=0.2$. Conditioning changes the denominator.
The area under the <survival function> up to a horizon gives the <restricted mean survival time>. A positive terminal plateau in a plotted <Kaplan–Meier estimator> does not establish that the occurrence will never happen: <terminal censoring and survival-mean identifiability> explains why the unobserved tail remains undetermined.
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