An introduction to geometrical probability: distributional aspects with applications
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam
Gordon and Breach
1999
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| Series: | Statistical distributions and models with applications
1 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 554 S. graph. Darst. |
| ISBN: | 9056996819 |
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| adam_text | 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
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 OF SOME 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 /C-DIMENSIONAL EUCLIDEAN SPACE 59 1.3.4 A
RANDOM PLANE IN A FC-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 OF INTERSECTIONS OF PAIRS OF
CHORDS OF A CONVEX FIGURE AND CROFTON S 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 OF AN 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 VII 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 GEOMETRICAL 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 N 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 OF THE P-CONTENT OF THE P-PARALLELOTOPE IN R N 340 3.2.3
SPHERICALLY SYMMETRIC DISTRIBUTION FOR X 341 3.2.4 THE DENSITY OF X AS A
FUNCTION OF THE ELEMENTS OF 5 = X X 343 3.2.5 THE DENSITY OF X AS A
FUNCTION OF X U, U U = I P 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 + 1)-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 2 367 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 OF RELATIVE VOLUMES 382 3.5.2 A
REPRESENTATION OF THE 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 RC-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 X CONTENTS 4.3
TYPE-1 BETA DISTRIBUTED RANDOM POINTS IN R N 422 4.3.1 SOME SPECIAL
CASES 425 EXERCISES 428 4.4 TYPE-2 BETA DISTRIBUTED RANDOM POINTS IN R N
430 4.4.1 SOME SPECIAL CASES 433 EXERCISES 436 4.5 GAUSSIAN DISTRIBUTED
RANDOM POINTS IN R N 437 4.5.1 THE DENSITY AS A G-FUNCTION 438 4.5.2
SOME SPECIAL CASES 438 EXERCISES 440 4.6 APPROXIMATIONS AND ASYMPTOTIC
RESULTS 441 4.6.1 APPROXIMATION IN THE CASE OF UNIFORMLY DISTRIBUTED
RANDOM POINTS 441 4.6.2 MILES CONJECTURE 444 4.6.3 APPROXIMATIONS IN
THE CASE OF TYPE-1 BETA DISTRIBUTED POINTS 445 4.6.4 APPROXIMATIONS IN
THE CASE OF TYPE-2 BETA DISTRIBUTED POINTS 446 4.6.5 ASYMPTOTIC RESULTS
FOR PRODUCT DISTRIBUTIONS 448 EXERCISES 449 4.7 MISCELLANEOUS RANDOM
VOLUMES AND THEIR DISTRIBUTIONS 453 4.7.1 PROBABILITY CONTENT OF CONES
IN RANDOM FIELDS 453 4.7.2 PROBABILITY CONTENT OF ELLIPTICAL CYLINDERS
AND ACID RAIN PROBLEM 454 4.7.3 VORONOI AND DELAUNAY TESSELLATIONS
GENERATED BY A STATIONARY POISSON POINT PROCESS 455 4.7.4 RANDOM CAPS ON
A SPHERE 459 EXERCISES 466 APPENDIX A*SOME STATISTICAL CONCEPTS 469 AL
JOINT DENSITY AND EXPECTED VALUES 469 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
|
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| discipline | Mathematik |
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| spelling | Mathai, Arakaparampil M. 1935- Verfasser (DE-588)141316241 aut An introduction to geometrical probability distributional aspects with applications A. M. Mathai Amsterdam Gordon and Breach 1999 XXII, 554 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Statistical distributions and models with applications 1 Geometric probabilities Random sets Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd rswk-swf Geometrische Wahrscheinlichkeit (DE-588)4156727-4 s DE-604 Statistical distributions and models with applications 1 (DE-604)BV013628478 1 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008645310&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mathai, Arakaparampil M. 1935- An introduction to geometrical probability distributional aspects with applications Statistical distributions and models with applications Geometric probabilities Random sets Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd |
| subject_GND | (DE-588)4156727-4 |
| title | An introduction to geometrical probability distributional aspects with applications |
| title_auth | An introduction to geometrical probability distributional aspects with applications |
| title_exact_search | An introduction to geometrical probability distributional aspects with applications |
| title_full | An introduction to geometrical probability distributional aspects with applications A. M. Mathai |
| title_fullStr | An introduction to geometrical probability distributional aspects with applications A. M. Mathai |
| title_full_unstemmed | An introduction to geometrical probability distributional aspects with applications A. M. Mathai |
| title_short | An introduction to geometrical probability |
| title_sort | an introduction to geometrical probability distributional aspects with applications |
| title_sub | distributional aspects with applications |
| topic | Geometric probabilities Random sets Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd |
| topic_facet | Geometric probabilities Random sets Geometrische Wahrscheinlichkeit |
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