 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/h/solution

The time already elapsed since the noise is

32

minutes. Conditional on a particular noise time

T = t

P (S - t - 32 > u ∣ S > t + 32) = e^{- u / \overset{s}{ˉ}}, u \geq 0.

This law does not depend on

t

. Averaging over the conditional distribution of

T

therefore leaves the same remaining-time law. Hence

E [m] = \overset{s}{ˉ} minutes .

The elapsed

32

minutes are not subtracted from the original mean; the exponential residual lifetime has the same mean after any elapsed duration.

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