An Introduction to Geometrical Probability

Transcription

1 An Introduction to Geometrical Probability Distributional Aspects with Applications A. M. Mathai McGill University Montreal, Canada Gordon and Breach Science Publishers Australia Canada China Prance Germany India Japan Luxembourg Malaysia The Netherlands Russia Singapore Switzerland

2 CONTENTS LIST OF TABLES LIST OF FIGURES ABOUT THE SERIES PREFACE xiii xv xix xxi 1. PRELIMINARIES INTRODUCTION BUFFON'S CLEAN TILE PROBLEM AND THE NEEDLE PROBLEM The Clean Tile problem The Needle Problem Buffon's Needle Problem With a Long Needle Long Needle and the Number of Cuts Buffon's Needle Problem With a Bent or Curved Needle The Grid Problem Coleman's Infinite Needle Problem Needle on Non-Rectangular Lattices 23 EXERCISES SOME GEOMETRICAL OBJECTS Regular Polyhedra n-dimensional Volume Contents and Surface Areas of Some Commonly Occurring Geometrical Objects Centre of Gravity of Plane Geometrical Objects Space Curve and Curvatures 48

3 vi CONTENTS EXERCISES PROBABILITY MEASURES AND INVARIANCE PROPERTIES A Random Line in a Plane in Cartesian Coordinates A Random Line in a Plane in Polar Coordinates A Random Plane in a /c-dimensional Euclidean Space A Random Plane in a fc-dimensional Euclidean Space in Polar Coordinates Infinitesimal Transformations A Measure for the Set of Lines in a Plane 72 EXERCISES MEASURES FOR POINTS OF INTERSECTION AND RANDOM ROTATIONS Density of Intersections of Pairs of Chords of a Convex Figure and Crofton's First Theorem on Convex Figures Crofton's Second Theorem on Convex Figures Density for Pairs of Points Random Division of a Plane Convex Figure by Lines A Measure for the Set of Planes Cutting a Line Segment Random Rotations The Kinematic Density for a Group of Motions in a Plane 107 EXERCISES RANDOM POINTS AND RANDOM DISTANCES INTRODUCTION RANDOM POINTS Random Points on a Line and the Random Division of an Interval Random Points by Poisson Arrivals Random Removal of Points from a Line 136 EXERCISES 141

4 CONTENTS vii 2.2 RANDOM DISTANCES ON A LINE AND SOME GENERAL PROCEDURES Random Points on a Line Segment Moments of a Random Line Segment A General Procedure Crofton's Theorem on Measures Crofton's Theorem on Mean Values Sylvester's Four Point Problem 159 EXERCISES RANDOM DISTANCES IN A CIRCLE Two Points on a Circle and Random Arcs Two Points on a Circle and Random Chords Bertrand's Paradox Distance Between a Fixed Point Outside and a Random Point Inside a Circle Distance Between Two Random Points Inside a Circle The Distance Between Random Points in Two Concentric Circles Distance Between Random Points in Nonoverlapping Circles 217 EXERCISES RANDOM POINTS IN A PLANE AND RANDOM POINTS IN RECTANGLES The Nearest Neighbor Problem on a Plane From Poisson Arrivals of Random Points Two Random Points Associated With a Rectangle Distance Between Random Points in Two Different Rectangles Other Types of Distances 246 EXERCISES RANDOM DISTANCES IN A TRIANGLE Random Points in a Triangle Distance of a Random Point in a Triangle From a Vertex 263 EXERCISES 274

5 viii CONTENTS 2.6 RANDOM DISTANCES IN A CONVEX BODY The Nearest Neighbor Problem Random Paths Across Convex Bodies, Stereological Probes Distance Between Two Random Points in a Hypersphere Distance Between Two Random Points in a Cube 296 EXERCISES RANDOM AREAS AND RANDOM VOLUMES GEOMETRICAL INTRODUCTION THE CONTENT OF A RANDOM PARALLELOTOPE The Distribution of the Content of a Random Parallelotope Random r-content of an r-simplex in R n Spherically Symmetric Case 333 EXERCISES RANDOM VOLUME, AN ALGEBRAIC PROCEDURE Some Results on Jacobians Distribution of the p-content of the p-parallelotope in R n Spherically Symmetric Distribution for X The Density of X as a Function of the Elements of 5 = X'X The Density of X as a Function of X'U, U'U = I p 345 EXERCISES RANDOM POINTS IN AN n-ball Uniformly Distributed Random Points in an n-ball Rotation Invariant (r + 1)-Figure Distributions Rotationally Invariant, Independently and Identically Distributed Random Points 360 EXERCISES 364

6 CONTENTS ix 3.4 CONVEX HULLS GENERATED BY RANDOM POINTS Convex Hull of p Points When the Dimension n = Convex Hull of p Points When the Dimension n > Convex Hull of Random Points in a Convex Body Convex Hull of Random Points in a Ball 376 EXERCISES RANDOM SIMPLEX IN A GIVEN SIMPLEX Invariance Properties of Relative Volumes A Representation of the Volume Content in Terms of Exponential Variables Random Triangle in a Given Triangle Moments of the Area of a Random Triangle Inside a Given Triangle 389 EXERCISES DISTRIBUTIONS OF RANDOM VOLUMES INTRODUCTION THE METHOD OF MOMENTS G- and H-Functions 397 EXERCISES UNIFORMLY DISTRIBUTED RANDOM POINTS IN A UNIT rc-ball Exact Density of the r-content as a G-Function Some Special Cases The Exact Density in Multiple Integrals for the General Case Exact Density in Multiple Series for the General Case Exact Density in Beta Series for the General Case 415 EXERCISES 421

7 x CONTENTS 4.3 TYPE-1 BETA DISTRIBUTED RANDOM POINTS IN R n Some Special Cases 425 EXERCISES TYPE-2 BETA DISTRIBUTED RANDOM POINTS IN R n Some Special Cases 433 EXERCISES GAUSSIAN DISTRIBUTED RANDOM POINTS IN R n The Density as a G-Function Some Special Cases 438 EXERCISES APPROXIMATIONS AND ASYMPTOTIC RESULTS Approximation in the Case of Uniformly Distributed Random Points Miles' Conjecture Approximations in the Case of Type-1 Beta Distributed Points Approximations in the Case of Type-2 Beta Distributed Points Asymptotic Results for Product Distributions 448 EXERCISES MISCELLANEOUS RANDOM VOLUMES AND THEIR DISTRIBUTIONS Probability Content of Cones in Random Fields Probability Content of Elliptical Cylinders and Acid Rain Problem Voronoi and Delaunay Tessellations Generated by a Stationary Poisson Point Process Random Caps on a Sphere 459 EXERCISES 466 Appendix A SOME STATISTICAL CONCEPTS 469 Al JOINT DENSITY AND EXPECTED VALUES 469

8 CONTENTS xi A2 REAL TYPE-1 AND TYPE-2 DIRICHLET DENSITIES 472 A3 HYPERGEOMETRIC SERIES 473 A4 LAURICELLA FUNCTION f D 475 A5 MELLIN TRANSFORM 475 Appendix B SOME REVISION MATERIAL FROM BASIC GEOMETRY 477 Bl DIRECTED LINE SEGMENT AND DIRECTION COSINES 477 B2 A DIRECTED LINE SEGMENT IN SPACE 479 B3 SOLID ANGLE 481 Appendix C SOME RESULTS FROM SPHERICALLY SYMMETRIC AND ELLIPTICALLY CONTOURED DISTRIBUTIONS 483 Cl SPHERICALLY SYMMETRIC DISTRIBUTIONS 483 C2 MULTIVARIATE GAUSSIAN DENSITY 484 C3 ELLIPTICALLY CONTOURED DISTRIBUTIONS 486 GLOSSARY OF SYMBOLS 487 BIBLIOGRAPHY 491 AUTHOR INDEX 539 SUBJECT INDEX 547