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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 27L / b / i / Solution

Write

r = (x^{2} + y^{2} + z^{2})^{1/2}

, so

f (x, y, z) = r^{3}

. For

t \geq 0

,

f^{- 1} ([0, t]) = {(x, y, z) : r \leq t^{1/3}},

whose volume is

\frac{4 π}{3} (t^{1/3})^{3} = \frac{4 π}{3} t .

Thus the pushforward intensity of

[0, t]

is

λ vol (f^{- 1} ([0, t])) = \frac{4 πλ}{3} t .

The pushforward is locally finite and diffuse, so the radial volume transform of a homogeneous Poisson point process and the mapping theorem show that

f (Π) is a homogeneous Poisson process of rate ρ = \frac{4 πλ}{3} .

Solved by gpt-5.6-sol high.

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