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An introduction to geometrical probability : distributional aspects with applications

Author: A M Mathai
Publisher: Amsterdam : Gordon & Breach, Science Pub., ©1999.
Series: Statistical distributions and models with applications, v. 1.
Edition/Format:   Print book : EnglishView all editions and formats
Database:WorldCat
Summary:

An in-depth look at applied and geometrical probability with an emphasis on statistical distributions, this text is a meticulous treatment of geometrical probability, kept at a level to appeal to a  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: A M Mathai
ISBN: 9056996819 9789056996819
OCLC Number: 43989534
Description: xxii, 554 pages : illustrations ; 24 cm.
Contents: 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 --
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 --
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 --
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 --
2. Random Points and Random Distances 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 --
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 --
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 --
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 --
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 --
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 --
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[superscript n] 330 --
3.1.3 Spherically Symmetric Case 333 --
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[superscript 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 S = X'X 343 --
3.2.5 The Density of X as a Function of X'U, U'U = I[subscript p] 345 --
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 --
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 [greater than or equal] 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 --
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 --
4. Distributions of Random Volumes 395 --
4.1 The Method of Moments 396 --
4.1.1 G- and H-Functions 397 --
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 --
4.3 Type-1 Beta Distributed Random Points In R[superscript n] 422 --
4.3.1 Some Special Cases 425 --
4.4 Type-2 Beta Distributed Random Points in R[superscript n] 430 --
4.4.1 Some Special Cases 433 --
4.5 Gaussian Distributed Random Points In R[superscript n] 437 --
4.5.1 The Density as a G-Function 438 --
4.5.2 Some Special Cases 438 --
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 --
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 --
Appendix A Some Statistical Concepts 469 --
A1 Joint Density and Expected Values 469 --
A2 Real Type-1 and Type-2 Dirichlet Densities 472 --
A3 Hypergeometric Series 473 --
A4 Lauricella Function f[subscript D] 475 --
A5 Mellin Transform 475 --
Appendix B Some Revision Material From Basic Geometry 477 --
B1 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 --
C1 Spherically Symmetric Distributions 483 --
C2 Multivariate Gaussian Density 484 --
C3 Elliptically Contoured Distributions 486.
Series Title: Statistical distributions and models with applications, v. 1.
Responsibility: A.M. Mathai.

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