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

Under the same independence assumption, inclusion-exclusion gives

R (t) = P_{1} (t) + P_{2} (t) - P (t) = ⎩ ⎨ ⎧ 0, s + s^{2} - s^{3}, 1, t \leq 0, 0 < t < T, t \geq T .

0 < t < T

R^{'} (t) = \frac{1 + 2 s - 3 s ^{2}}{T} = \frac{( 1 - s ) ( 3 s + 1 )}{T} > 0, R^{''} (t) = \frac{2 - 6 s}{T ^{2}} .

There is one interior inflection point, where the curvature changes from positive to negative:

(t, R (t)) = (\frac{T}{3}, \frac{11}{27}) .

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

s \leq R (t) \leq 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.

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