2.1 The survival function

Let \(T \geq 0\) be a survival time with distribution function \(F(t) = P(T \leq t)\) and density \(f(t) = F'(t)\).

\(S(0) = 1\), \(S\) is non-increasing, and \(S(t) \to 0\) as \(t \to \infty \) (if everyone eventually has the event). Its graph, the survival curve, shows the proportion still event-free at each time; estimating it from censored data is the subject of the section on the Kaplan–Meier estimator. The probability of surviving from \(a\) to \(b\) is \(S(a) - S(b)\), and the probability of surviving a further \(u\), having survived to \(t\), is the conditional survival \[ P(T > t + u \mid T > t) = \frac {S(t + u)}{S(t)} . \]

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