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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 18H / a / Solution

Write

X_{i} = θ + U_{i}

, where the

U_{i}

are independent

Uniform (0, 1)

variables. Convolution gives the sum of two independent uniform variables:

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) .

Solved by gpt-5.6-sol high.

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