Consider a Poisson random scatter of points in a plane with mean intensity \(\lambda\) per unit area. Let \(R\) be the distance from zero to the closest point of the scatter.
- Find a formula for the c.d.f. and the density of \(R\) and sketch their graphs.
- Show that \(\sqrt{2 \lambda \pi} R\) has the Rayleigh distribution.
- Find the mean and mode of \(R\).
[pitman p 389, # 21]