= Hazard function
{title2=$h(t)$}
{wiki}
= Hazard
{synonym}
For a <survival time> with an <absolutely continuous probability distribution>, <probability density function> $f$, and <survival function> $S$, the <hazard function> is
$$
h(t)=\frac{f(t)}{S(t)}=-\frac{S'(t)}{S(t)}
$$
where $S(t)>0$, with the <derivative> identity holding <almost everywhere>. At points where the local limit exists, equivalently
$$
h(t)=\lim_{\Delta\downarrow0}\frac{\mathbb P(t<T\leq t+\Delta\mid T>t)}{\Delta}.
$$
The <hazard function> measures an instantaneous conditional rate. If time is in years, its units are inverse years, and its value can exceed one. A <probability> for a short <interval> is approximately $h(t)\Delta$, not $h(t)$; for an <absolutely continuous probability distribution>, the <probability> of an occurrence at the single exact time $t$ is zero. The <discrete hazard>, by contrast, is a conditional <probability> at a possible event time.
The factorization $f(t)=h(t)S(t)$ distinguishes a <probability density function> for the original population from a <hazard function> for those still event-free. With an <exponential distribution> of rate $0.2$ per year, $h(5)=0.2$ per year but $f(5)=0.2e^{-1}\approx0.073576$ per year: the latter also accounts for the <probability> of reaching year five. The <exponential survival risk over an interval> example computes finite-horizon <probabilities> from this rate.
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