Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 14 / f / Solution

Under the same independence assumption, inclusion-exclusion givesOn $0<t<T$,There is one interior inflection point, where the curvature changes from positive to negative:At $t=T$ the graph reaches one with zero slope and then stays constant. There is no further change from one sign of curvature to the opposite sign there.

Without independence, $R=P_1+P_2-P$ still holds, but only $\boxed{s\le R(t)\le \min (1,s+s^2)}$ is forced by the marginals. Its inflection behaviour is then not determined. The shared-$U$ construction in (e), for example, gives $R=s$ before the plateau.