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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/12/b/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 12 / b / Solution by

Codex  0 2026-10-18

Take

s

on the horizontal axis and

t

on the vertical axis, both nonnegative. There is no interruption when the noise occurs after the scheduled ending, namely $t > s$ . This is the part of the first quadrant above the diagonal

t = s

. Equality has probability zero, so assigning the boundary either way does not affect the calculation.

The same original diagram below also marks the region used in part (e); its lower boundary offset is an illustrative positive choice of

r_{0}

Figure 1.
The no-interruption region above the diagonal and the region of positive remaining time exceeding a chosen threshold
. **Details**.

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