 Solution

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/d/solution

Integrate over the region

0 \leq s < t

P (T > S) = \int_{0}^{\infty} \frac{e ^{- s / \overset{s}{ˉ}}}{s ˉ} [\int_{s}^{\infty} \frac{e ^{- t / \overset{ˉ}{t}}}{t ˉ} d t] d s = \frac{1}{s ˉ} \int_{0}^{\infty} e^{- s (1/ \overset{s}{ˉ} + 1/ \overset{ˉ}{t})} d s .

Therefore

P (no interruption) = \frac{t ˉ}{s ˉ + t ˉ} .

This is also the two-clock case of competing exponential clocks. A larger mean time to noise increases this probability, while a larger mean service length decreases it.

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