If discs of radius have centers randomly distributed over the unit square, how much of the square will be covered? What is the underlying probability distribution function?

Stochastic coverage processes have medical, military, and natural science applications: respectively, the overlapping disks represent antibodies connected to a virus, the destruction from a bomb drop, and the coverage of a tree canopy. This Demonstration uses the torus convention, where each disk that protrudes from one side of the square is considered to enter the opposite side. This convention simplifies the mathematics, but its effect decreases with . The left graphic shows a sample run for a given and, while the right graphic plots the current coverage overlaid on the mean and variance curves for this value.

Equations for the mean and variance are given in [1].

Reference

[1] P. G. Hall, Introduction to the Theory of Coverage Processes, New York: Wiley Series in Probability and Mathematical Statistics, 1988 pp. 24–25.