 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-3/29k/b/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

On the interior of the support

b < m

f_{B_{t}, M_{t}} (b, m) = \frac{2 ( 2 m - b )}{2 π t ^{3/2}} exp (- \frac{( 2 m - b ) ^{2}}{2 t}) .

y = m

and

z = m - b

. Its inverse is

m = y

b = y - z

2 m - b = y + z

, for

y, z \geq 0

this yields

f_{M_{t}, Z_{t}} (y, z) = \frac{2 ( y + z )}{2 π t t} e^{- (y + z)^{2} / (2 t)} .

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