 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-48/5/b/solution

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

The event that the first arrival exceeds

t

is exactly the event

N_{t} = 0

. Thus

P (τ_{1} > t) = e^{- λ t}

for

t \geq 0

f_{τ_{1}} (t) = λ e^{- λ t}

for

t > 0

E τ_{1} = \int_{0}^{\infty} e^{- λ t} d t = 1/ λ

and

E τ_{1}^{2} = 2 \int_{0}^{\infty} t e^{- λ t} d t = 2/ λ^{2}

. Therefore $E τ_{1} = 1/ λ, Var τ_{1} = 1/ λ^{2}$ .

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