 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-33/5/a/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

For a continuous positive survival time,

S^{'} (t) = - f (t)

h (t) = f (t) / S (t)

wherever

S (t) > 0

. Therefore

\frac{d}{d t} lo g S (t) = - h (t) .

Since

S (0) = 1

, integrating gives the cumulative hazard function identity

S (t) = exp {- H (t)}, H (t) = \int_{0}^{t} h (u) d u .

At an endpoint where survival reaches zero, the corresponding cumulative hazard is interpreted as

+ \infty

and the exponential as zero.

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