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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-207/6/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 207 / 6 / d / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

By conditional probability and continuity of event times,

P (J = A ∣ T < τ) = \frac{F _{A} ( τ )}{F _{A} ( τ ) + F _{B} ( τ )} .

When

τ > 0

and

q = θ_{A} + θ_{B} > 0

, the common factor

1 - e^{- q τ}

cancels, giving

P (J = A ∣ T < τ) = \frac{θ _{A}}{θ _{A} + θ _{B}} .

Before

τ

, the constant cause-specific hazards divide the total event intensity in these proportions, regardless of how long the interval is. If

τ = 0

or both intensities are zero, the conditioning event has probability zero and this conditional probability is undefined, not a ratio assigned by convention.

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