Intersections of Random Convex Regions.
STANFORD UNIV CALIF DEPT OF STATISTICS
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By a random region is meant a region whose size and shape are fixed but whose orientation and location are random variables. The joint distribution of these random variables is assumed invariant under translations but not necessarily under rotations of the coordinate axes. Mean values are obtained describing the intersection of a random convex region with a fixed convex region, namely, the mean area and perimeter of the intersection of such regions on the plane and the mean volume and surface area of the intersection of such regions in three dimensions. These mean values are generalized to provide the mean area, volume, etc. of the intersection within a fixed convex region of n independent random convex regions which intersect the fixed region. Similar mean values, applicable to coverages problems, are obtained for the union within a fixed convex region of n independent random convex regions which intersect the fixed region. Modified author abstract
- Statistics and Probability