With common fixed censoring time $c$, the ordering is known exactly when the earlier event occurs by $c$; continuity makes ties have probability zero. Thus the informative event is $\{min(T_1,T_2)\leq c\}$ and
$$
\mathbb P(T_1<T_2\mid\text{informative})
=\frac{\int_0^c f_1(t)F_2(t)\,dt}
{1-F_1(c)F_2(c)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 5 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-207/5/a/ii/solution