Mean-Value Formulae for a Class of Random Sets

Abstract

Summary Mean‐value formulae for important characteristics (e.g. volume, surface area, etc.) of the common intersection of randomly placed point‐sets are derived under the condition that the basic probability distribution of the underlying model is either motion‐invariant or translation‐invariant (the orientation of the generating sets in this latter case being known). Various special cases are discussed, some of them leading to a new derivation of known results of geometrical probability theory. The expected value of the probability that an arbitrary point is covered by the random intersection‐set is also determined.

A fourth note on recent research in geometrical probability

“…The random positioning is with respect to the invariant measure under Euclidean motion or under translations only (so in the latter the domains have fixed orientation). For the one, two, and three-dimensional cases, Streit (1973) derives formulae for the mean values of the fundamental characteristics (volume, surface etc.) of the intersection K; n K 1 n • .. n Ki; These lead to some well-known results of geometrical probability (some of which are stated by Miles above) ano to moments of the length of a random secant and mean probabilities of coverage (using Robbins' theorem).…”