Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 4 / a / ii

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- Solution ii

Solution

Under the null hypothesis of equal event-time distributions, every member of the combined risk set has the same instantaneous chance of being the next event. Conditional on one event at

a_{j}

and on the two risk-set sizes,

U_{j} \sim Bernoulli (\frac{r _{j}^{(1)}}{r _{j}^{(0)} + r _{j}^{(1)}}),

so

E_{0} (U_{j} ∣ r_{j}^{(0)}, r_{j}^{(1)}) = \frac{r _{j}^{(1)}}{r _{j}^{(0)} + r _{j}^{(1)}} .

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