An Introduction to Geometrical Probability - Distributional Aspects with Application

579 Pages·1999·21.479 MB·English

by  A. M. Mathai

An Introduction
to Geometrical Probability
Distributional Aspects with Applications
A. M~ Mathai
McGill University
Montreal, Canada
Gordon and Breach Science Publishers
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Copyright @1999 OPA (Overseas Publishers Association) N. V. Published
by license under the Gordon and Breach Science Publishers imprint.
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British Library Cataloguing in Publication Data
Mathai, A. M.
An introduction to geometrical probability: distributional
aspects with applications. - (Statistical distributions and
models with applications; v. 1 - ISSN 1028-8929)
1. Geometric probabilities 2. Random sets
I. Title
519.2
ISBN90-5699-681-9
CONTENTS
LIST OF TABLES xiii
LIST OF FIGURES xv
ABOUT THE SERIES xix
PREFACE xxi
1. PRELIMINARIES 1
1.0 INTRODUCTION 1
1.1 BUFFON'S CLEAN TILE PROBLEM AND
THE NEEDLE PROBLEM 1
1.1.1 The Clean Tile problem 2
1.1.2 The Needle Problem 5
1.1.3 Buffon's Needle Problem With a Long Needle 9
1.1.4 Long Needle and the Number of Cuts 12
1.1.5 Buffon's Needle Problem With a Bent
or Curved Needle 14
1.1.6 The Grid Problem 17
1.1.7 Coleman's Infinite Needle Problem 20
1.1.8 Needle on Non-Rectangular Lattices 23
EXERCISES 26
1.2 SOME GEOMETRICAL OBJECTS 30
1.2.1 Regular Polyhedra 30
1.2.2 n-Dimensional Volume Contents and Surface
Areas ofSome Commonly Occurring
Geometrical Objects 31
1.2.3 Centre of Gravity of Plane Geometrical Objects 46
1.2.4 Space Curve and Curvatures 48
vi CONTENTS
EXERCISES 50
1.3 PROBABILITY MEASURES AND INVARIANCE
PROPERTIES 52
1.3.1 A Random Line in a Plane in Cartesian
Coordinates 55
1.3.2 A Random Line in a Plane in Polar Coordinates 58
1.3.3 A Random Plane in a k-Dimensional Euclidean
Space 59
1.3.4 A Random Plane in a k-Dimensional Euclidean
Space in Polar Coordinates 62
1.3.5 Infinitesimal Transformations 64
1.3.6 A Measure for the Set of Lines in a Plane 72
EXERCISES 83
1.4 MEASURES FOR POINTS OF INTERSECTION
AND RANDOM ROTATIONS 86
1.4.1 Density ofIntersections of Pairs of Chords of a
Convex Figure and CroftOD'8 First Theorem
on Convex Figures 86
1.4.2 Crofton's Second Theorem on Convex Figures 89
1.4.3 Density for Pairs of Points 91
1.4.4 Random Division of a Plane Convex Figure
by Lines 95
1.4.5 A Measure for the Set of Planes Cutting a Line
Segment 98
1.4.6 Random Rotations 104
1.4.7 The Kinematic Density for a Group of Motions
in a Plane 107
EXERCISES 108
2. RANDOM POINTS AND RANDOM DISTANCES 111
2.0 INTRODUCTION 111
2.1 RANDOM POINTS 111
2.1.1 Random Points on a Line and the Random
Division ofan Interval 113
2.1.2 Random Points by Poisson Arrivals 128
2.1.3 Random Removal of Points from a Line 136
EXERCISES 141
CONTENTS vU
2.2 RANDOM DISTANCES ON A LINE AND
SOME GENERAL PROCEDURES 145
2.2.1 Random Points on a Line Segment 145
2.2.2 Moments of a Random Line Segment 147
2.2.3 A General Procedure 150
2.2.4 Crofton's Theorem on Measures 152
2.2.5 Crofton's Theorem on Mean Values 156
2.2.6 Sylvester's Four Point Problem 159
EXERCISES 168
2.3 RANDOM DISTANCES IN A CIRCLE 171
2.3.1 Two Points on a Circle and Random Arcs 171
2.3.2 Two Points on a Circle and Random Chords 172
2.3.3 Bertrand's Paradox 179
2.3.4 Distance Between a Fixed Point Outside and a
Random Point Inside a Circle 187
2.3.5 Distance Between Two Random Points Inside a
Circle 203
2.3.6 The Distance Between Random Points in Two
Concentric Circles 211
2.3.7 Distance Between Random Points in
Nonoverlapping Circles 217
EXERCISES 220
2.4 RANDOM POINTS IN A PLANE AND RANDOM
POINTS IN RECTANGLES 223
2.4.1 The Nearest Neighbor Problem on a Plane
From Poisson Arrivals of Random Points 223
2.4.2 Two Random Points Associated With a
Rectangle 228
2.4.3 Distance Between Random Points in Two
Different Rectangles 241
2.4.4 Other Types of Distances 246
EXERCISES 259
2.5 RANDOM DISTANCES IN A TRIANGLE 262
2.5.1 Random Points in a Triangle 262
2.5.2 Distance of a Random Point in a Triangle
From a Vertex 263
EXERCISES 274
viii CONTENTS
2.6 RANDOM DISTANCES IN A CONVEX BODY 276
2.6.1 The Nearest Neighbor Problem 276
2.6.2 Random Paths Across Convex Bodies,
Stereological Probes 278
2.6.3 Distance Between Two Random Points in a
Hypersphere 289
2.6.4 Distance Between Two Random Points in a
Cube 296
EXERCISES 310
3. RANDOM AREAS AND RANDOM VOLUMES 315
3.0 GEOMETRlCAL INTRODUCTION 315
3.1 THE CONTENT OF A RANDOM
PARALLELOTOPE 323
3.1.1 The Distribution of the Content of a
Random Parallelotope 324
3.1.2 Random r-Content of an r-Simplex in R" 330
3.1.3 Spherically Symmetric Case 333
EXERCISES 335
3.2 RANDOM VOLUME, AN ALGEBRAIC
PROCEDURE 336
3.2.1 Some Results on Jacobians 336
3.2.2 Distribution ofthe p-Content of the
p-Parallelotope in Rn 340
3.2.3 Spherically Symmetric Distribution for X 341
3.2.4 The Density ofX as a Function ofthe
Elements of S == X'X 343
3.2.5 The Density of X as a Function of
X'U, U'U:::::: t, 345
EXERCISES 350
3.3 RANDOM POINTS IN AN n-BALL 353
3.3.1 Uniformly Distributed Random Points in an
n-Ball 353
3.3.2 Rotation Invariant (r+I)-Figure Distributions 358
3.3.3 Rotationally Invariant, Independently and
Identically Distributed Random Points 360
EXERCISES 364
CONTENTS ix
3.4 CONVEX HULLS GENERATED BY RANDOM
POINTS 366
3.4.1 Convex Hull of p Points When the
Dimension n == 1 366
3.4.2 Convex Hull of p Points When the
Dimension n 2:: 2 367
3.4.3 Convex Hull of Random Points in a Convex
Body 371
3.4.4 Convex Hull of Random Points in a Ball 376
EXERCISES 381
3.5 RANDOM SIMPLEX IN A GIVEN SIMPLEX 382
3.5.1 Invariance Properties ofRelative Volumes 382
3.5.2 A Representation ofthe Volume Content
in Terms of Exponential Variables 383
3.5.3 Random Triangle in a Given Triangle 386
3.5.4 Moments of the Area of a Random Triangle
Inside a Given Triangle 389
EXERCISES 392
4. DISTRIBUTIONS OF RANDOM VOLUMES 395
4.0 INTRODUCTION 395
4~1 THE METHOD OF MOMENTS 396
4.1.1 G- and H-Functions 397
EXERCISES 401
4.2 UNIFORMLY DISTRIBUTED RANDOM POINTS
IN A UNIT n-BALL 403
4.2.1 Exact Density of the r-Content as a
G-Function 404
4.2~2 Some Special Cases 406
4.2.3 The Exact Density in Multiple Integrals
for the General Case 410
4.2.4 Exact Density in Multiple Series for
the General Case 413
4.2.5 Exact Density in Beta Series for the
General Case 415
EXERCISES 421