A Generalization of the Integral of the Circular Coverage Function

Abstract

where V is the volume within S2. We seek the probability that the sphere S1 will contain X' and will show that it is expressible in terms of integrals of the noncentral chi-square distribution. The result lends itself to several interpretations. First, summarizing the previous paragraph, it may be regarded as the probability that a sphere S1 of radius R, whose center has an n-dimensional standard normal distribution, will contain a point which is distributed uniformly inside of or on a sphere S2 of radius D centered at the origin. Secondly, it is the probability that a sphere Si of radius R contains a randomly selected point in a sphere S2 of radius D after Si reaches its destination if the center of Si is thrown at the center of S2 and the point at which the center of Si comes to rest is spherically normally distributed. Finally, it may be interpreted as the expected fraction of a sphere S2 of radius D lying within a sphere SI of radius R when Si reaches its destination if the center of Si is thrown at the center of S2 and the point at which the center of SI comes to rest is spherically normally distributed.