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?
Radius of a hole.
-1
$\begingroup$
statistics
expectation
uniform-distribution
maximum-likelihood
-
0It is _not possible_ to have "uniformly distributed points on a plane". What you intended to be the second sentence isn't a sentence. There is a moderately large collection of paradoxes in geometrical probability that start with this faulty premise. (Perhaps be best known one is to find the probability that three such "uniformly distributed" points form an obtuse triangle.) – 2017-01-09
1 Answers
0
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.