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On random circles in a square and on a circle. (English. Ukrainian original) Zbl 0959.60004
Theory Probab. Math. Stat. 59, 109-112 (1999); translation from Teor. Jmorvirn. Mat. Stat. 59, 106-109 (1998).
F. Garwood [Biometrika 34, 1-17 (1947; Zbl 0030.20402)] considered the following problem: let the centers of \(N\) disks with radius \(a>0\) be independently and uniformly distributed over a domain \(T\) formed by points of the plane. The distance between these points and the points of unit square \(A\) does not exceed \(a\). It is necessary to find: (1) the probability that the point \(M\in A\) is covered by at least one disk, and (2) the probability that two points \(M_{1}, M_{2}\in A\) are not covered by any disk. The set \(T\) is a sum (by Minkowski) of the square \(A\) and a disk of radius \(a\). The answer to the first question is \(1-[(1+4a)/(1+4a+\pi a^{2})]^{4}\). The answer to the second question is more compound and this probability is expressed by some integral \(I(N).\) The aim of this paper is to find the asymptotic behavior of the integral \(I(N)\) as \(N\to+\infty\) and to find the “principal part” of this integral.

60D05 Geometric probability and stochastic geometry
60E10 Characteristic functions; other transforms
60C05 Combinatorial probability