This book is intended as a reference for use by researchers in the fields
of Probability and Statistics. It may also serve as supplementary reading
for students who have already taken a good calculus-based course on Probability
Theory. A working knowledge of multivariable calculus is necessary
for following the intricate mathematical developments that are presented in
this book. Since any rigorous discussion on geometrical probability would
require the reader to have a sophisticated mathematical background (like
measure theory and integral geometry), the discussion in this book is kept
at a semi-rigorous and introductory level, so that the exposition will appeal
to a wider audience including applied researchers. The level and nature of
the discussion is kept at a pace which will make the book suitable for selfstudy.
In addition, each section includes several illustrative examples with
additional results being presented as exercises. This should facilitate the
use of the book as a text for a special topics course either at the senior
undergraduate or graduate level.
Applied researchers will find this book to be both functional and practical
through the use of a large number of problems chosen from different
disciplines. Geometrical probabilities encompass a large number of topics
which would entail a very lengthy discussion even at an introductory level.
The author, therefore, has carefully chosen some of the important topics
which would appeal to the general readership. A few topics such as packing
and covering problems which have a vast literature are introduced here at
a peripheral level for the purpose of familiarizing readers who are new to
this area of research.
Author(s): A. M. Mathai
Series: Statistical Distributions and Models with Applications - Volume 1
Publisher: Gordon and Breach Science Publishers
Year: 1999
Language: English
Pages: XXII; 554
Title Page
Contents
LIST OF TABLES
LIST OF FIGURES
ABOUT THE SERIES
PREFACE
1. PRELIMINARIES
1.0 INTRODUCTION
1.1 BUFFON'S CLEAN TILE PROBLEM AND THE NEEDLE PROBLEM
1.2 SOME GEOMETRICAL OBJECTS
1.3 PROBABILITY MEASURES AND INVARIANCE PROPERTIES
1.4 MEASURES FOR POINTS OF INTERSECTION AND RANDOM ROTATIONS
2. RANDOM POINTS AND RANDOM DISTANCES
2.0 INTRODUCTION
2.1 RANDOM POINTS
2.2 RANDOM DISTANCES ON A LINE AND SOME GENERAL PROCEDURES
2.3 RANDOM DISTANCES IN A CIRCLE
2.4 RANDOM POINTS IN A PLANE AND RANDOM POINTS IN RECTANGLES
2.5 RANDOM DISTANCES IN A TRIANGLE
2.6 RANDOM DISTANCES IN A CONVEX BODY
3. RANDOM AREAS AND RANDOM VOLUMES
3.0 GEOMETRlCAL INTRODUCTION
3.1 THE CONTENT OF A RANDOM PARALLELOTOPE
3.2 RANDOM VOLUME, AN ALGEBRAIC PROCEDURE
3.3 RANDOM POINTS IN AN n-BALL
3.4 CONVEX HULLS GENERATED BY RANDOMPOINTS
3.5 RANDOM SIMPLEX IN A GIVEN SIMPLEX
4. DISTRIBUTIONS OF RANDOM VOLUMES
4.0 INTRODUCTION
4.1 THE METHOD OF MOMENTS
4.2 UNIFORMLY DISTRIBUTED RANDOM POINTS IN A UNIT n-BALL
4.3 TYPE-l BETA DISTRIBUTED RANDOM POINTS IN R[n]
4.4 TYPE-2 BETA DISTRIBUTED RANDOM POINTS IN R[n]
4.5 GAUSSIAN DISTRIBUTED RANDOM POINTS IN R[n]
4.6 APPROXIMATIONS AND ASYMPTOTIC RESULTS
4.7 MISCELLANEOUS RANDOM VOLUMES AND THEIR DISTRlBUTIONS
Appendix A - SOME STATISTICAL CONCEPTS
A1 - JOINT DENSITY AND EXPECTED VALUES
A2 - REAL TYPE-l AND TYPE-2 DIRICHLET DENSITIES
A3 - HYPERGEOMETRIC SERIES
A4 - LAURlCELLA FUNCTION f[D]
A5 - MELLIN TRANSFORM
Appendix B - SOME REVISION MATERIAL FROM BASIC GEOMETRY
B1 - DIRECTED LINE SEGMENT AND DIRECTION COSINES
B2 - A DIRECTED LINE SEGMENT IN SPACE
B3 - SOLID ANGLE
Appendix C - SOME RESULTS FROM SPHERICALLY SYMMETRIC AND ELLIPTICALLY CONTOURED DISTRIBUTIONS
C1 - SPHERICALLY SYMMETRIC DISTRIBUTIONS
C2 - MULTIVARIATE GAUSSIAN DENSITY
C3 - ELLIPTICALLY CONTOURED DISTRlBUTIONS
GLOSSARY OF SYMBOLS
BIBLIOGRAPHY
AUTHOR INDEX
SUBJECT INDEX
REAR COVER
