Geometric ProbabilitySIAM, 1978年1月1日 - 180 頁 Topics include: ways modern statistical procedures can yield estimates of pi more precisely than the original Buffon procedure traditionally used; the question of density and measure for random geometric elements that leave probability and expectation statements invariant under translation and rotation; the number of random line intersections in a plane and their angles of intersection; developments due to W.L. Stevens's ingenious solution for evaluating the probability that n random arcs of size a cover a unit circumference completely; the development of M.W. Crofton's mean value theorem and its applications in classical problems; and an interesting problem in geometrical probability presented by a karyograph. |
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angle Applications approximately arcs assume average axis boundary cell circle circumference completely component condition connected consider contained convex region coordinates coverage covered crossings curve define denote density derive developed dimensions direction distance distribution divided E(Area endpoint equal equation estimator evaluate event example expected expression falls find first fixed formed four fragments function geometrical given grid half Hence hits independent inside integral invariant length line segment lines intersecting mean measure method moments namely needle normal Note number of intersections obtain origin pair parallel perimeter perpendicular placed plane polygon positions probability problem projection radius random chord random lines random points random variable region respectively result shown side specified sphere statistic Suppose surface taken triangle uniformly distributed values volume