An Introduction to Geometrical Probability - Distributional Aspects with Application 9056996819

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An Introduction to Geometrical Probability Distributional Aspects with Applications

A.

M~

Mathai

McGill University Montreal, Canada

Gordon and Breach Science Publishers Australia Japan

Canada China France Germany India Luxembourg Malaysia The Netherlands Russia

Singapore

Switzerland

Copyright @1999 OPA (Overseas Publishers Association) N. V. Published by license under the Gordon and Breach Science Publishers imprint. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the publisher. Printed in Singapore.

Amsteldijk 166 1st Floor 1079 LH Amsterdam The Netherlands

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

ISBN 90-5699-681-9

CONTENTS LIST OF TABLES LIST OF FIGURES

xiii xv

ABOUT THE SERIES

xix

PREFACE

xxi

1. PRELIMINARIES 1.0 INTRODUCTION 1.1 BUFFON'S CLEAN TILE PROBLEM AND THE NEEDLE PROBLEM 1.1.1 The Clean Tile problem 1.1.2 The Needle Problem 1.1.3 Buffon's Needle Problem With a Long Needle 1.1.4 Long Needle and the Number of Cuts 1.1.5 Buffon's Needle Problem With a Bent or Curved Needle 1.1.6 The Grid Problem 1.1.7 Coleman's Infinite Needle Problem 1.1.8 Needle on Non-Rectangular Lattices

EXERCISES 1.2 SOME GEOMETRICAL OBJECTS 1.2.1 Regular Polyhedra 1.2.2 n-Dimensional Volume Contents and Surface Areas of Some Commonly Occurring Geometrical Objects 1.2.3 Centre of Gravity of Plane Geometrical Objects 1.2.4 Space Curve and Curvatures

1 1 1 2 5 9 12

14 17 20 23 26 30 30

31 46 48

CONTENTS

vi

EXERCISES 1.3 PROBABILITY MEASURES AND INVARIANCE PROPERTIES 1.3.1 A Random Line in a Plane in Cartesian Coordinates 1.3.2 A Random Line in a Plane in Polar Coordinates 1.3.3 A Random Plane in a k-Dimensional Euclidean Space 1.3.4 A Random Plane in a k-Dimensional Euclidean Space in Polar Coordinates 1.3.5 Infinitesimal Transformations 1.3.6 A Measure for the Set of Lines in a Plane

EXERCISES 1.4 MEASURES FOR POINTS OF INTERSECTION AND RANDOM ROTATIONS

50 52 55 58 59 62 64 72

83 86

1.4.1 Density of Intersections of Pairs of Chords of a Convex Figure and Croft OD '8 First Theorem on Convex Figures 1.4.2 Crofton's Second Theorem on Convex Figures 1.4.3 Density for Pairs of Points 1.4.4 Random Division of a Plane Convex Figure by Lines 1.4.5 A Measure for the Set of Planes Cutting a Line Segment 1.4.6 Random Rotations 1.4.7 The Kinematic Density for a Group of Motions in a Plane

107

EXERCISES

108

2. RANDOM POINTS AND RANDOM DISTANCES

86 89 91

95 98 104

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 2.1.2 Random Points by Poisson Arrivals 2.1.3 Random Removal of Points from a Line

113 128 136

EXERCISES

141

CONTENTS 2.2 RANDOM DISTANCES ON A LINE AND SOME GENERAL PROCEDURES 2.2.1 Random Points on a Line Segment 2.2.2 Moments of a Random Line Segment 2.2.3 A General Procedure 2.2.4 Crofton's Theorem on Measures 2.2.5 Crofton's Theorem on Mean Values 2.2.6 Sylvester's Four Point Problem EXERCISES 2.3 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 2.3.5 Distance Between Two Random Points Inside a Circle 2.3.6 The Distance Between Random Points in Two Concentric Circles 2.3.7 Distance Between Random Points in Nonoverlapping Circles 2.3.1 2.3.2 2.3.3 2.3.4

EXERCISES 2.4 RANDOM POINTS IN A PLANE AND RANDOM POINTS IN RECTANGLES 2.4.1 The Nearest Neighbor Problem on a Plane From Poisson Arrivals of Random Points 2.4.2 Two Random Points Associated With a Rectangle 2.4.3 Distance Between Random Points in Two Different Rectangles 2.4.4 Other Types of Distances EXERCISES 2.5 RANDOM DISTANCES IN A TRIANGLE 2.5.1 Random Points in a Triangle 2.5.2 Distance of a Random Point in a Triangle From a Vertex EXERCISES

vU

145 145 147 150 152 156 159 168 171 171 172 179

187 203 211

217 220 223 223 228

241 246 259 262 262 263 274

CONTENTS

viii

2.6 RANDOM DISTANCES IN A CONVEX BODY 2.6.1 The Nearest Neighbor Problem 2.6.2 Random Paths Across Convex Bodies, Stereological Probes 2.6.3 Distance Between Two Random Points in a Hypersphere 2.6.4 Distance Between Two Random Points in a Cube EXERCISES

3. RANDOM AREAS AND RANDOM VOLUMES 3.0 GEOMETRlCAL INTRODUCTION

3.1 THE CONTENT OF A RANDOM PARALLELOTOPE 3.1.1 The Distribution of the Content of a Random Parallelotope 3.1.2 Random r-Content of an r-Simplex in R" 3.1.3 Spherically Symmetric Case EXERCISES 3.2 RANDOM VOLUME, AN ALGEBRAIC PROCEDURE

276 276

278 289 296

310

315 315

323 324 330 333

335 336

3.2.1 Some Results on Jacobians 3.2.2 Distribution of the p-Content of the p-Parallelotope in Rn 3.2.3 Spherically Symmetric Distribution for X 3.2.4 The Density of X as a Function of the Elements of S == X' X 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 3.3.1 Uniformly Distributed Random Points in an n-Ball 3.3.2 Rotation Invariant (r + I)-Figure Distributions 3.3.3 Rotationally Invariant, Independently and Identically Distributed Random Points

EXERCISES

336 340 341

343

353 353 358 360

364

CONTENTS 3.4 CONVEX HULLS GENERATED BY RANDOM POINTS 3.4.1 Convex Hull Dimension n 3.4.2 Convex Hull Dimension n 3.4.3 Convex Hull Body 3.4.4 Convex Hull

of p Points When the 1 of p Points When the 2:: 2 of Random Points in a Convex

==

of Random Points in a Ball

EXERCISES 3.5 RANDOM SIMPLEX IN A GIVEN SIMPLEX

ix

366 366

367

371 376 381

382

3.5.1 Invariance Properties of Relative Volumes 3.5.2 A Representation of the Volume Content in Terms of Exponential Variables 3.5.3 Random Triangle in a Given Triangle 3.5.4 Moments of the Area of a Random Triangle Inside a Given Triangle

389

EXERCISES

392

4. DISTRIBUTIONS OF RANDOM VOLUMES

382

383 386

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 4.2~2 Some Special Cases 4.2.3 The Exact Density in Multiple Integrals for the General Case 4.2.4 Exact Density in Multiple Series for the General Case 4.2.5 Exact Density in Beta Series for the General Case

415

EXERCISES

421

404 406 410

413

CONTENTS

x

4.3 TYPE-l BETA DISTRIBUTED RANDOM POINTS IN s:

422

4.3.1 Some Special Cases

425

EXERCISES

428

4.4 TYPE-2 BETA DISTRIBUTED RANDOM POINTS IN Rn

430

4.4.1 Some Special Cases

433

EXERCISES

436

4.5 GAUSSIAN DISTRIBUTED RANDOM POINTS IN Rn

437

4.5.1 The Density as a G-F\mction 4.5.2 Some Special Cases

438 438

EXERCISES

440

4.6 APPROXIMATIONS AND ASYMPTOTIC RESULTS

441

4.6.1 Approximation in the Case of Uniformly Distributed Random Points 4.6.2 Miles' Conjecture 4.6.3 Approximations in the Case of Type-l Beta Distributed Points 4.6.4 Approximations in the Case of Type-2 Beta Distributed Points 4.6.5 Asymptotic Results for Product Distributions

446 448

EXERCISES

449

4.7 MISCELLANEOUS RANDOM VOLUMES AND THEIR DISTRlBUTIONS 4.7.1 Probability Content of Cones in Random Fields 4.7.2 Probability Content of Elliptical Cylinders and Acid Rain Problem 4.7.3 Voronoi and Delaunay Tessellations Generated by a Stationary Poisson Point Process 4.7.4 Random Caps on a Sphere

EXERCISES Appendix A-SOME STATISTICAL CONCEPTS Al JOINT DENSITY AND EXPECTED VALUES

441 444 445

453 453 454 455 459 466

469 469

CONTENTS

~

A2 REAL TYPE-l AND TYPE-2 DIRICHLET DENSITIES

472

A3 HYPERGEOMETRIC SERIES

473

A4 LAURlCELLA FUNCTION ID

475

A5 MELLIN TRANSFORM

475

Appendix B-BOME 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 DISTRlBUTIONS

486

GLOSSARY OF SYMBOLS

487

BIBLIOGRAPHY

491

AUTHOR INDEX

539

SUBJECT INDEX

547

PRELIMINAlliES

5

Another case considered by Buffon is a tile shaped a hexagon of side l and the circular coin of diameter d. For the coin to fall clean, the centre of the coin must be within the inner hexagon as shown in Figure 1.1.4.

FIGURE 1.1.4.

Circular coin on a hexagon

The area of the hexagon of side 1 is A hexagon is

=

~ J3l2 . The area of the inner

a=~[~l-~r· Then the probability of no cut is game,

~

and proceeding as before, for a fair

which gives the admissible solution l

=

~ [1 + /2]d ~ 1.971197d.

(1.1.4)

BuffOD also looked into the odds for each of the other players and examined the relationship between land d for the game to be fair. In a certain way this clean tile problem of Buffon represents the first attempt towards computing probabilities by using geometry instead of analysis. But this is not the whole story! Buffon goes even further and enquires about a more challenging question, the famous needle problem. 1.1.2

The Needle Problem

A straight needle or a headless pin of length l is dropped at random on a set of equidistant parallel lines in the plane. [By referring to the previous

PRELIMINARIES

1.3.1

55

A Random Line in a Plane in Cartesian Coordinates

A line in a two-dimensional Euclidean space can be represented in different ways. In Cartesian coordinates (x, y) an arbitrary line, not passing through the origin, can be given by the equation

ux + vy + 1 = 0

(1.3.4)

where u and v are parameters. This line is called a random line when u and v are random variables. For example, 2x + y + 1 =: 0 is one such line where u takes the value 2 and v takes the value 1. A line passing through the origin is obtained from (1.3.4) by translation, that is , by relocating the origin. The effect of rotating the axes through an angle Q is shown in Figure 1.3.3(b). The line is ux +vy+ 1 = 0 in (x, y)-axes and u·x· +v*y* + 1 == 0 in (x·,y*)-axes. Note that x

O B , BC = y. tan Q => cOSQ'

X

x*

=:

OB+BC= --+y*tanQ :::}

x

=

x*cOSCt-y·sinQ.

COSQ

Similarly y == x" sin a

+ y. COSQ

or

x" =

X

COSQ

+Y

sin Cl:' and y.

= -x

sin a

+ y cos o.

If the origin is also shifted to a new position, say, x to x + a and y to y + b, then

x· y*

x cos a: + y sin a: + a COSQ + b sin o, -x sino+y coso+b cosa-a sin o,

and then

o= u·x* + v·y* + 1 =>

116

RANDOM POINTS AND RANDOM DISTANCES

the (VI, ... , vn-k)-th product moment of tl, ... , tn-k is given by

where n = {(t1, ...,fn-l)/O < tj < 1, j = 1, ... ,n -1, L,;~;tj < I} and L = L(tl, ... , t n - 1 ) can also be written as

Hence from the type--l Dirichlet integral (see Appenclix A2) we have the following:

Theorem 2.1.2

E

For t l , ... , tn-l as defined in Lemma 2.1.2

k) (til1 l .•• tllnn-k

f( n )

1

+ 1t Vl 1

n xt~~i+l

1

tlln-k+l-l

... n-k

t~:\ (1 - t l - ... - tn_l)l-ldtl ... dtn_l

fev! + l) r(vn - k + l)r(n) r(Vl + + Vn-k + n) for rR(Vj)

> -1,

j

=:

1, ..., n - k, where

~(.)

denotes the real part of (.).

If we consider positive integer moments then f(vj

+ 1) =

Vj!.

Now for all

11,

(2.1.9)

since 0

< u < 1, where for example, (a)m = a(a + l) ...(a + m-I), (a)o = 1, a i- O.

118

RANDOM POINTS AND R.L\.NDOM DISTANCES n

IT

=

[1 ~ Orn+l-jtJ- 1

j=k+l

which when substituted for =

aj

is

IT n

[

1-

(j - k) t .

J

-1

(2.1.11)

J

j=k+l

for ItI < 1. Replace t by t~l in (2.1.11) and write v:::::: 1- u. Then (2.1.11) red uces to the following:

Theorem 2.1.4 that ItI < ~J

For v as defined in (2.1.7) and for an arbitrary t such 1-

vt) -n]

E [ ( 1=t which implies for

[j ~ J= Jll kt

n

1

j(l - t)

It! < !' ,

E[(l-vt)-n] = ~;(l-t)~k

n

IT

(j-kt)-l.

(2.1.13)

i=k+l

Note that (2.1.13), on expansion, can also generate the moments of v. What is the range of v for the nonzero part of the density? This can be seen from the range of tl, ... , t n . If t 1 , •.. , t.; are the n Cartesian coordinates then the point (tl t n ) lies inside the (n - I)-dimensional tetrahedron T whose n vertices are (1,0, ... ,0), (0, 1,0, ... ,0), ... , (0, ... ,0,1). Then tl, ... , t« together with the condition u :::::: E;~t ajtj imply that the density of u is proportional to the hyperarea of the (n - 2)-dimensional convex rectilinear figure obtained by the intersection of T with the hyperplane E j ajtj == u. This hyperplane passes through a vertex of T when u = aj and hence aj > U > aj+ 1 which, when the aj'8 are substituted for, implies that t ••• ,

k

k.

- - - 0 and !j(Xj) = 0 elsewhere. One method of deriving the density of x is to use transformation of variables on successive XIX2. The joint density of Xl and X2 is fl(Xl)!2(X2). products. Let Ul Then making the transformation Ul = Xl X2, t = Xl, the Jacobian is t- 1 J and substituting in the joint density of Xl and X2, the density of Ul J denoted by 9l(Ul), is easily seen to be the following:

=

91 (UI)

== D2ur2-111 tOl-02-02(1 - t)Ol- 1(t - Ut}°2- 1dt

(ii)

'Ul

where (iii)

Making the substitution 1 91(Ul)

== D2U f 2- 1 (1 -

1 1

x

tl

== (t -

Ul) /

(1 -

Ul)

we have

Ul)t'31+,62- 1

tfl- 1 (1 -

td 02- 1 [1 -

tt{1 - udJ-(02+02-0Ildt1'

(iv)

Now take U2 = UlXa and apply the above procedure. Successive applications of the above transformations give the final result. If I(x) denotes the density of x = Xl ••• X2r then we have t The exact density of x in (4.2.6), denoted by f(x), is

Theorem 4.2.4 given by

f(x)

=

D2rXQ2r-l(1 - X)02r-l 1 X

[

10

IT t1,-1 (1 -

1 { 2r-l

, •• [

10

ti).8i+ 1 -

1

i=l

x[1- t 2r- 1t2r-2 ...t2r- i(1 - x)t"'Y2r-idti}. 0 < x < 1 and f(x)

= 0 elsewhere, di =

f3l

where D2r is given in (iii) above,

+ ... + f3i

and 'Yi = Qi+l

+ {3i+l -

O'i·

(v)

438 4.5.1

DISTRIBUTIONS OF RANDOlvI VOLUf\1ES The Density as a G-Function

By examining the moment expression in (4~5.1) it is evident that the density of ~n can be written as an H-function. From (4.5.3) it is apparent that the density of x can be written as a G-function.

Theorem 4.5.1 given by

The exact density of x in (4.5.2), denoted by f(x)., is

where

Logarithmic series expansion of such a G-function, for general parameters, is available in Mathai and Saxena (1973) and in earlier papers of Mathai. Some particular cases can be written down in terms of elementary functions. 4.5.2

Some Special Cases

From the gamma structure in (4.5.3) one special case is evident that when = 1, x has a real gamma density given by

T

x]--le- x

f(x)=

Case 2:

In this case

r(Jf)

,O 0 is a real symmetric positive definite matrix and c > 0 is a fixed real number. The trajectory of this ellipsoid forrns an elliptical cylinder. Then the volume of air swept out by such a cloud can be approximated by an elliptical cylinder and thus the amount of pollutants absorbed by the cloud is proportional to the probability content of this elliptical cylinder in a 3-dimensional Gaussian field. Without loss of generality one can describe this elliptical cylinder as follows: Consider an offset ellipse in the variables Xl and X2, perpendicular to the x3-axis, and moving from X3 == b1 to Xa :::::: bz where b, and bz are fixed numbers. The offset ellipse can be described as

(4.7.3) where ai, a2, A are known numbers and B =: B ' > 0 is a 2 x 2 real symmetric positive definite, and known, matrix. The probability content of this is available from the conditional density of (Xl, X2) given X3, denoted by fl((xl,x2)lx3). The probability content is

lh

«Xl. X2) IX 3)dx 1 dX2

DISTRIBUTIONS OF RANDOM VOLlJMES

457

Then the k-th moment of u has the following form:

r(~+k)r(!!.f-+k) [r (~+ k)]n+l

c------~,..;;;.......-~

{nj=o r [2(~$~) + nh + k]}

x-~-------------~

{nj:~ r (~ + ~ + k) }

(4.7.7)

where c is the normalizing constant such that the right side is unity when k == O. It is easy to see that E(u k ) exists for k > -~ when n == 1 and k

> -;(~1g for

n

~ 2.

It is not difficult to show that E(u h ) exists for a

complex variable h such that ~(h) > - ~(~1~~ for n ~ 1. Hence the moment expression with k replaced by h can be directly inverted (take the inverse Mellin transform) to obtain the density of u. One can also examine the· moment sequence in (4.7.7) for k = 0,1, ... to see that a set of sufficient conditions for the unique determination of the density of u~ through this moment sequence, are satisfied by (4.7.7). Some special cases.

n = 1.

Case 1:

In this case the right side of (4.7.7) reduces to the following: c

r (~ + k) rn + k) = 1T-~r (~ + k) rn + k).

(4.7.8)

Replacing k by k/2 and applying the duplication formula for gamma functions, (4.7.8) reduces to 2- k r (1 + k). Substituting back for V n one has

which is the k-th moment of an exponential variable. Hence w = PVl is an exponential variable with the density e- w , W > o. Case 2:

n = 2.

DISTRIBUTIONS OF R.A.NDOrvI VOLUMES

463

A lower bound can be derived by. a similar argument. This is left as an exercise at the en cl of this section. For the general 0: and N it is difficult to evaluate p{o)(N). The first two moments of z, the proportion of the total surface area not covered by the N random caps, will be derived here. Let I~rv(Q) be the indicator function of the event that the point Q is not covered by the N random caps. That is, IN(Q) = {1 if Q is ~ot covered by IV caps o otherwise,

Then x

=~ 47f

J

IN(Q)dQ

where dQ is the element of the surface area around Q. Evidently, for N == 1,

where A is given in

(4.7.19)~

and hence

Then E(x)

= :=

4~

J

E[IN(Q)dQ

(4.7.25)

E[IN(Q)] == (1 - A)N.

Let us evaluate the second moment of x .

where Q1 and Q2 are two different points with the elements of the areas dQl and dQ2 respectively. Let the points Ql and Q2 be separated by an angle () and let 1(8) denote the combined surface area covered by the caps of the same angular radius Q centred at Ql and Q2 respectively. l\. random cap covers one of the points Q1 and Q2 if the pole of the cap falls on the region with area f«()). Then the probability that a random cap covers one of the points is The probability that N random caps leave both Ql and Q2 uncovered is

/1:).

464

DISTRIBUTIONS OF RANDOM VOLUIvIES

Then

JJ (4~)2 JJ[1- f1:)]N (4~)2

E[IN(Ql)IN(Q2))dQ 1dQ2

dQ 1dQ2.

The angle subtended by QI and Q2 is denoted by B. The two points Q1 and Q2 subtend an angle between and () + dO at the centre of the sphere if and only if the second point falls in the ring of area 27f sin 0 dO (see also Figure 4.7.1 (c)). This event happens with probability

e

27r sin 0 dO _ ~ · () dO 41T - 2 SIn • Substituting this in E(x 2 ) we have

E(x2 )

=

~ 2

r

la

[1-

f(B)]N sin 0 dO. 47r

(4.7.26)

Note that if () 2 2Q then the two caps with poles at Q1 and Q2 do not overlap. Then from (4.7.18) 1(0)

== 2[27r(1 - cos Q)] = 471(1 - cos a)

or

1 - 1(0) == COSQ. 411' If () < 2Q then the covered regions by the two caps intersect. The union of the two covered regions can be computed with the help of spherical trigonometry, and the final result is the following:

o ~ 7r cosa.[l- ~COS-l (::!)] COSll,

f(O) 1--= 47r

20

~

+ 17T CO8-1 (s~n!) sIn e r '

0

< () -