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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/2005/paper-1/4/b/i/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 / 2005 / Paper 1 / 4 / b / i / Solution by

Codex  0 Created 2026-10-18 Updated 2026-10-20

For

s, τ \geq 0

, use conditional probability and the survival function:

P (T > s + τ ∣ T > s) = \frac{e ^{- α (s + τ)}}{e ^{- α s}} = e^{- ατ} .

It is independent of

s

and equals

P (T > τ)

. This is the memoryless property of the exponential distribution, equivalent here to a constant hazard function

α

.

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