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]

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} .

ii

Solution

Put

T_{k} = R_{k}^{3}

. Part (i) identifies

T_{k}

as the

k

th arrival time of a rate

ρ = \frac{4 πλ}{3}

T_{k}

f_{T_{k}} (t) = \frac{ρ ^{k}}{( k - 1 )!} t^{k - 1} e^{- ρt}, t > 0.

t = r^{3}

, with

d t = 3 r^{2} d r

f_{R_{k}} (r) = \frac{3 ρ ^{k}}{( k - 1 )!} r^{3 k - 1} e^{- ρ r^{3}}, r > 0, ρ = \frac{4 πλ}{3} .

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