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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.

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