Solution

Let

N

be the homogeneous Poisson point process in

R^{3}

with intensity measure

λ d x

, and map each point

x

to its radius

g (x) = ∣ x ∣

. For an interval

(a, b] \subseteq R_{+}

, the pushforward measure of the intensity is

λ vol {x : a < ∣ x ∣ \leq b} = \frac{4 πλ}{3} (b^{3} - a^{3}) .

The Mapping theorem for Poisson point processes therefore shows that the radii form a Non-homogeneous Poisson point process with intensity function

ρ (r) = \frac{d}{d r} (\frac{4 πλ r ^{3}}{3}) = 4 πλ r^{2}, r > 0.

R

is the distance to the closest star, the event

{R > r}

says that the ball of radius

r

contains no points. The zero-count probability of a Poisson distribution gives

P (R > r) = exp (- \frac{4 πλ r ^{3}}{3}) .

f_{R} (r) = 4 πλ r^{2} exp (- \frac{4 πλ r ^{3}}{3}), r \geq 0.

Solved by gpt-5.6-sol high.

 Ancestors (11)