Wahrscheinlichkeitsrate

Wahrscheinlichkeitsrate is a term used in statistics to denote the instantaneous rate at which a given event occurs, expressed as probability per unit time. It is not a single universally standardized concept, but is closely related to the hazard rate in survival analysis and to the intensity function in stochastic processes. In many contexts the idea is to quantify how likely an event is to happen at an exact moment, given that it has not happened yet.

In continuous-time models, let T be the time of the event with probability density f(t) and survival function S(t) = P(T > t) = 1 − F(t). The hazard rate h(t) is defined as h(t) = f(t) / S(t) = (dF/dt) / (1 − F(t)). It represents the conditional instantaneous probability: the probability that the event occurs in the next infinitesimal interval dt given survival up to time t, approximately h(t) dt. The cumulative hazard is H(t) = ∫0^t h(u) du, and F(t) = 1 − exp(−H(t)) in several common models (e.g., exponential). For non-homogeneous Poisson processes, the rate function λ(t) serves a similar role, with P(event in [t, t+dt) | no event before t) ≈ λ(t) dt.

In discrete time or informal usage, the Wahrscheinlichkeitsrate may refer to the probability of the event within a small interval or to a model parameter that governs the frequency of events (such as λ in a Poisson process).

Terminology varies by field: German literature often uses Gefährdungsrate (hazard rate) or Intensitätsfunktion for the same idea. The term Wahrscheinlichkeitsrate should be defined clearly from context.