 Waiting-time distribution

A waiting-time distribution is the probability distribution of a nonnegative random duration $T$ until a specified event. It can be discrete, absolutely continuous, or mixed. Its survival function is $S(t)=\mathbb{P}(T>t)$ and its cumulative distribution function is $F(t)=1-S(t)$.

If the finite-time law is absolutely continuous, the probability density function is $f(t)=-S^{\prime }(t)$ almost everywhere. Mere continuity of the cumulative distribution function is insufficient for this density representation. At times with positive survival, the hazard function is $h(t)=f(t)/S(t)$; it measures the instantaneous event rate conditional on having waited that long.

Constant hazard $h(t)=\lambda >0$ with $S(0)=1$ gives $S(t)=e^{-\lambda t}$, the exponential distribution. Waiting a discrete number of independent Bernoulli trials instead gives a geometric distribution. A fixed deterministic wait has a point mass at its prescribed duration rather than an ordinary density.

Sequential independent waiting stages add, so their probability distributions combine by convolution. The two-step catalytic waiting time gives a concrete chemical-kinetics example. Waiting-time distributions also describe first-passage times and time-to-event observations in survival analysis.