Distribution of the Measure of a Random Two-dimensional Set

Abstract

This paper considers the distribution of the measure of a special random two-dimensional set. Related work, usually motivated by a search for principles for bombing operations, deals exclusively with moment problems and appears in [1], [3], [4], [5], [6], [7], [8]. A one-dimensional distribution problem appears in [2]. The random set considered is the intersection of a fixed circle with the union of $N$ random circles. Centers of the random circles are subject to the variability imposed by the bivariate normal distribution with circular symmetry and means not necessarily coincident with the coordinates of the center of the fixed circle. The measure of interest is the ratio of the area of the intersection ("covered area") to the total area of the fixed circle. For $N = 1,$ the distribution is determined and its use facilitated by the graphs in Fig. 1 and Fig. 2. A procedure for obtaining upper and lower bounds of the distribution for $N = 2$ is given. Tables I, II, III, and IV give upper and lower bounds for the percentage points of the distribution for $N = 2$ for some special illustrative situations. For $N = 1$ in all situations, and for $N = 2$ in many situations; the graphs and tables demonstrate that a realistic decision can be made rather easily without resorting to the usual practice of random number "Monte Carlo" devices for each ad hoc situation of interest.