 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-1/18h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Write

X_{i} = θ + U_{i}

, where the

U_{i}

are independent

Uniform (0, 1)

f_{X_{1} + X_{2}} (s) = ⎩ ⎨ ⎧ s - 2 θ, 2 θ + 2 - s, 0, 2 θ \leq s \leq 2 θ + 1, 2 θ + 1 \leq s \leq 2 θ + 2, otherwise .

Its plot is a triangle with vertices

(2 θ, 0), (2 θ + 1, 1), (2 θ + 2, 0) .

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