 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2010/m/m4/a/ii/solution

Integrate the joint density over

u

f_{V} (v) = \int_{- \infty}^{\infty} ∣ u ∣ f (u) g (uv) d u .

Since both original densities are even, substitution

u \mapsto - u

shows that the negative and positive halves contribute equally. Hence

f_{V} (v) = 2 \int_{0}^{\infty} u f (u) g (uv) d u .

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