 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/16/a/v/solution

The atom at zero contributes zero to the expected value, while all positive delays must retain their unconditional density. Therefore

μ = E [S] = 0 (1 - \frac{r}{R}) + \int_{0}^{r} \frac{s}{R} d s = \frac{r ^{2}}{2 R} .

This mean is

r / R

r /2

conditional on encountering a stop; using

r /2

alone would omit the vehicles passing without delay.

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