 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ia/paper-2/4f/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-05

y

to obtain the marginal density of

X

f_{X} (x) = \int_{0}^{\infty} x e^{- x (y + 1)} d y = e^{- x}, x > 0.

Thus

X

f_{Y ∣ X = x} (y) = x e^{- x y} 1_{{y \geq 0}}, x > 0.

Equivalently,

Y ∣ X = x

x

. The value at

x = 0

can be assigned arbitrarily: it is a probability-zero conditioning value and the density ratio there is not meaningful.

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