Distance to the nearest point

Topic: Poisson processes · more questions on this topic
‹ Three events in two hours

A homogeneous planar Poisson process has intensity 2 points per square kilometre. Find the median distance from a fixed location to the nearest point.

Worked answer (click to show)

The void probability is \mathbb{P}(R>r)=e^{-2\pi r^2}. The median solves \mathbb{P}(R\le r)=1/2, so e^{-2\pi r^2}=1/2. Then 2\pi r^2=\log 2 and

r=\sqrt{\frac{\log 2}{2\pi}}

kilometres.