= Waiting-time distribution
{title2=$p(t)=-S'(t)$}
A <waiting-time distribution> is the <probability distribution> of a nonnegative <random variable>[random duration] $T$ until a specified <event>. It can be discrete, <absolute continuity of measures>[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 <absolute continuity of measures>[absolutely continuous], the <probability density function> is $f(t)=-S'(t)$ <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 probability>[conditional] on having waited that long.
Constant <hazard function>[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 trial>[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>[chemical-kinetics] example. Waiting-time distributions also describe <first-passage time>[first-passage times] and time-to-event observations in <survival analysis>.
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