 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-35/7/5/solution

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

Divide the joint first-event/type-A density by the type probability. For

α > 0

f_{T ∣ J = A} (t) = \frac{α e ^{- (α + β) t}}{α / ( α + β )} = (α + β) e^{- (α + β) t}, t \geq 0.

It integrates to one, so the conditional time is exponential with rate

α + β

. Type-A events that win the competition tend to occur sooner than an unrestricted latent A time. In fact the same conditional density holds for B, so the observed first-event time and its type are independent in this constant-rate model.

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