SurvivalFunktion

SurvivalFunktion, often called the survival function, is a function used in probability and statistics to describe the distribution of a nonnegative random lifetime. For a random lifetime T, the survival function is defined as S(t) = P(T > t), equivalently S(t) = 1 − F(t), where F is the cumulative distribution function of T. The function is typically defined for t ≥ 0 and describes the probability that an individual or item survives beyond time t.

Key properties of the survival function include that it is nonincreasing and right-continuous, with S(0) = 1 (if T is almost surely nonnegative) and S(t) → 0 as t → ∞ for lifetimes that end almost surely. The derivative, when it exists, relates to the probability density function f(t) by S′(t) = −f(t). The hazard function h(t) = f(t)/S(t) connects the instantaneous failure rate to the remaining survival probability and is a central concept in reliability and survival analysis.

The survival function also provides useful representations of moments. For nonnegative lifetimes, the expected value E[T] can be expressed as E[T] = ∫0^∞ S(t) dt. Relationships to the CDF, moments, and tail behavior are used in modeling and inference.

Common examples include the exponential distribution, where S(t) = e^{−λt}, and the Weibull distribution, with S(t) = exp(−(t/α)^β). In practice, S(t) is estimated from data (often with censoring) using methods such as the Kaplan–Meier estimator or parametric/semi-parametric models. Applications span medical survival analysis, reliability engineering, and product lifetime studies.