Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 207 / 1 / f

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- Solution f

Solution

Integrating the constant density

2

across horizontal slices of the triangular support gives

f_{π_{1}} (x) = 2 (1 - x), 0 < x < 1,

so

π_{1} \sim Beta (1, 2)

. For

z = π_{2} = π_{1} + λ

, the line segment at fixed

z

z

, and the transformation has unit Jacobian. Hence

f_{π_{2}} (z) = 2 z, 0 < z < 1,

so

π_{1} \sim Beta (1, 2), π_{2} \sim Beta (2, 1) .

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