# Coverage of a Unit Square by Random Discs

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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?

Contributed by: Aaron Becker (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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.

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