 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-35/7/1/solution

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

Let

T = min (T_{A}, T_{B})

be the first event time and

J

its type. The cause-specific hazard for A is

h_{A} (t) = lim_{u ↓ 0} \frac{P r ( t \leq T < t + u , J = A ∣ T \geq t )}{u} .

It is a rate among people free of both event types, not the marginal probability of eventually experiencing A. Under the given independent latent event times, their constant hazard functions give exponential survivor functions. Therefore the first-event survivor is

S (t) = Pr (T_{A} \geq t, T_{B} \geq t) = e^{- α t} e^{- βt} = e^{- (α + β) t} .

The combined first-event rate is

s = α + β > 0

. Subsequent formulas concerning conditioning on type A require

α > 0

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