Geometric Probability (CBMS-NSF Regional Conference Series by Herbert Solomon

By Herbert Solomon

Themes comprise: methods sleek statistical approaches can yield estimates of pi extra accurately than the unique Buffon process routinely used; the query of density and degree for random geometric parts that go away likelihood and expectation statements invariant lower than translation and rotation; the variety of random line intersections in a aircraft and their angles of intersection; advancements as a result of W. L. Stevens's creative answer for comparing the likelihood that n random arcs of measurement a canopy a unit circumference thoroughly; the advance of M. W. Crofton's suggest worth theorem and its purposes in classical difficulties; and an enticing challenge in geometrical chance provided through a karyograph.

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Example text

Then dy = tan 0sec 6d6 = y ^ y 2 - 1 d# and the density of y is dy/(yVy 2 - i). Thus FIG. 3 The number of random lines intersecting disjoint sections of a line are clearly independent. Thus the points of intersection of the random lines with any arbitrary line constitute a Poisson process of intensity IT/TT. This result without the development appears in Miles (1964). Previously we derived some of the statistical properties of the polygons into which the random lines divide the plane. When the number of random lines is finite or the area of the plane under consideration is finite, difficulties arise with RANDOM LINES IN THE PLANE AND APPLICATIONS 47 edge effects and infinities.

See Fig. ) FIG. 7 24 CHAPTER 1 The probability that the line with endpoints P and P' forms an angle 0 with the radius of the sphere is the probability that the endpoint P' falls in the indicated circumferential belt or surface area of circumferential belt Probability = — — —— , surface area of sphere with radius r but 5 = rd implies and thus the desired density element CHAPTER 2 Density and Measure for Random Geometric Elements In our Buffon discussion we have referred to the random positioning of a line segment in the plane and in a brief way to the similar situation for a line in the plane.

For the present we turn our attention to the more general question of density and measure for random geometric elements that leave probability and expectation statements invariant under translation and rotation. To do this we borrow heavily from Santalo (1953), (1976). Remarkable results due to Crofton (1885) fall out of the analysis as the principal applications for this chapter. When a point in the plane is described by its Cartesian coordinates x, y from an arbitrary origin, the appropriate measure for the set of points in a region A is given by JA dx dy and this measure is invariant under translation and rotation.

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