Suppose there are points uniformly distributed on a plane and we are looking at this plane through a circular hole. Suppose we the points with the coordinates $\{x_i, y_i\}_{i=1}^{N}$. How can we find the radius of the hole as an expectation of the random variable and using the maximum likelihood principle?
2026-03-28 03:02:31.1774666951
Radius of a hole.
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Intuitively, you have some density of points per unit area. The expected number to see is that density times the area of the hole, so count the points you see, divide by the density to compute the area they represent, and convert that to a radius.
More carefully, you can think of each small area having a probability of having a point that is the density times the area. The distribution of the number of points seen will be Poisson with a mean that is the expected number of points to see.