Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 5 / b

For independent exponential distributions, the numerator from part a(ii) iswhile the denominator is the expression in braces. Their ratio is

Condition on the independently generated random censoring information. For every realized common censoring horizon, part b(i) gives the same conditional probability ${\lambda }_1/({\lambda }_1+{\lambda }_2)$. The law of total probability therefore gives that ratio after averaging over the censoring distribution as well.