Little Stochastic Geometry Problem: Random Circles

Little Stochastic Geometry Problem: Random Circles


This should remind you of your probability classes during college years. Can you solve this problem in 30 minutes? This could make for an interesting job interview question.
Problem
Points are randomly distributed on the plane, with an average of m points per unit area. A circle of radius R is drawn around each point. What is the proportion of the plane covered by these (possibly overlapping) circles? 
What if circles can have two different radii, either R = r, or R = r’, with same probability? What if R is a random variable, so that we are dealing with random circles? Before reading further, try to solve the problem yourself.

Solution
The points are distributed according to a Poisson point process of intensity m. The chance that an arbitrary point x in the plane is not covered by any circle, is the chance that there is zero point from the process, in a circle of radius R centered at x. This is equal to exp(-2mpR). Thus the proportion of area covered by the circles is p(m,R) = 1 –  exp(-2mpR). 
If circles have radii equal to r or r’, it is like having two Poisson processes, one for each type of circle. The chance that x is not covered by any circle is thus equal to  (exp(-2mpr) + exp(-2mpr’))/2. If R is a continuous random variable and f denotes the density, the general solution is

You can easily simulate a large number of these circles over a broad area, and then, pick up 1,000 random points and see how many of them are covered by at least one circle, to check whether your solution is correct or not. 
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Link: Little Stochastic Geometry Problem: Random Circles