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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 9F / i / Solution

Put

Q = \sum_{j = 1}^{n} q_{j}

. Since the variables have continuous distributions, the minimum is unique almost surely. Integrating over its possible value gives

P (K = k, T \geq t) = \int_{t}^{\infty} q_{k} e^{- q_{k} s} j \neq = k \prod P (S_{j} > s) d s = \int_{t}^{\infty} q_{k} e^{- Q s} d s = \frac{q _{k}}{Q} e^{- Qt} .

This is the joint tail calculation for competing exponential clocks.

Solved by gpt-5.6-sol high.

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