 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-101/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

For

t \geq 0

, independence factors the survival function of the minimum:

P (M > t) = \prod_{j = 1}^{n} P (X_{j} > t) = \prod_{j = 1}^{n} e^{- λ t} = e^{- nλ t} .

The minimum is nonnegative, and this is exactly the survival function of an exponential distribution. Hence

M \sim Exp (nλ) .

The parameter here is the rate, rather than the mean. The calculation is the simplest example of competing exponential clocks.

 New to topics? Read the docs here!