A two-dimensional Poisson process is a process of randomly occurring events in the plane such that (i) for any region of area A the number of events in that region has a Poisson distribution with mean...

A two-dimensional Poisson process is a process of randomly occurring events in the plane such that

(i) for any region of area A the number of events in that region has a Poisson distribution with mean λA, and

(ii) the number of events in non-overlapping regions are independent.

For such a process, consider an arbitrary point in the plane and let X denote its distance from its nearest event (where distance is measured in the usual Euclidean manner). Show that

(a) P{X>t} = e^−λπt2