Survival from the hazard

Topic: Reliability · more questions on this topic
‹ Mean life as an integral of survival

Let T have density f, survival function R and hazard h=f/R. Show that R(t)=\exp\bigl(-\int_0^t h(u)\,\,\mathrm{d} u\bigr) for t\ge 0.

Worked answer (click to show)

Since R=1-F, R'(t)=-f(t), so

h(t)=\frac{f(t)}{R(t)}=-\frac{R'(t)}{R(t)}=-\frac{\,\mathrm{d}}{\,\mathrm{d} t}\log R(t).

Integrating from 0 to t and using R(0)=\mathbb{P}(T>0)=1 for a lifetime with a density,

\log R(t)=-\int_0^th(u)\,\,\mathrm{d} u,\qquad R(t)=\exp\Bigl(-\int_0^th(u)\,\,\mathrm{d} u\Bigr).