reliabilitetsfunktion

The reliabilitetsfunktion, also known as the reliability function, denotes the probability that a component or system will function without failure up to a given time t. It is commonly written as S(t) and is defined in terms of the lifetime T of the item as S(t) = P(T > t). Equivalently, if F(t) = P(T ≤ t) is the cumulative distribution function of the lifetime, then S(t) = 1 − F(t).

The function S(t) is nonincreasing, with S(0) representing the probability of surviving from time zero, typically equal to 1 for a functioning start, and S(t) → 0 as t → ∞ for nonlifelong components. The expected lifetime can be obtained from S(t) by E[T] = ∫0^∞ S(t) dt when the integral exists.

Reliability theory links S(t) to other common functions in survival analysis. The density f(t) = dF/dt satisfies f(t) = −dS(t)/dt, and the hazard function h(t) = f(t)/S(t) describes the instantaneous failure rate. The cumulative hazard is H(t) = ∫0^t h(u) du, and S(t) = exp[−H(t)] in the usual continuous models. These relationships allow a unified treatment of lifetime data.

Estimation and application are central to reliability analysis. Nonparametric methods such as the Kaplan–Meier estimator estimate S(t) under censoring, while parametric models (exponential, Weibull, lognormal) provide structure for extrapolation and maintenance planning. The reliabilitetsfunktion is used to assess durability, schedule preventive maintenance, and compare reliability across designs.