Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 4

If $U\sim \mathrm{Uniform}(0,1)$, the inverse transform sampling formulahas survival function $\mathbb{P}(T>t)=e^{-\lambda t}$ and hence the required exponential distribution.

Without censoring, every $v_i=1$ and $\mathbb{E}{T}_i=1/\lambda$. ThereforeThis is events divided by total person-time at risk.

The first two mechanisms are standard independent, noninformative censoring. The third relabels the event time itself and therefore is not independent censoring for the original latent event time; it is more naturally interpreted as assigning a cause in a competing risks model. The detailed calculations follow in the three subparts.

Fixed administrative censoring $C=\tau$ is noninformative because it is unrelated to the random event time. The censored fraction isMoreover, $\mathbb{E}v=1-q$ andso the event-to-time ratio remains $\lambda$.

For an independent $C\sim \mathrm{Exponential}(\mu )$, censoring is noninformative andThe observed time is exponential with rate $\lambda +\mu$, while $\mathbb{E}v=\lambda /(\lambda +\mu )=1-q$. Thus

Independent relabelling gives $\pi =q$, $\mathbb{E}v=1-q$, and leaves $X=T$ with mean $1/\lambda$. Its ratio is therefore $\lambda (1-q)$ rather than $\lambda$. To retain event hazard $\lambda$, simulate the common occurrence time with rateand label it as the target event with probability $1-q$. This is equivalent to independent competing clocks with target-event rate $\lambda$ and censoring-cause rate $q\lambda /(1-q)$; their minimum has rate $\lambda /(1-q)$.