Topics In Probability

This book serves as an introduction catalogue to some of the many facets of geometrical and other patterns. It is informal, and the intended audience spans several fields. The book might be used by students, graphic artists, illustrators, scientists, artists, laypeople, programmers or anyone fascinated by optically provocative art. "The Pattern Book" combines old and new ideas - with emphasis on the fun that the true pattern lover finds in doing, rather than in reading about the doing! The book is organized into three main parts: Representing Nature (for those patterns which describe or show real physical phenomena, eg. visualizations of protein motion, jellyfish, etc), Mathematics and Symmetry (for those patterns which describe or show mathematical behaviour, eg. fractals), and Human Art (for those patterns which are artistic works of humans and made without the aid of a computer eg. Moslem tiling patterns). A comprehensive glossary is included to help ease readers into technical or unfamiliar waters Recent research in probability has been concerned with applications such as data mining and finance models. Some aspects of the foundations of probability theory have receded into the background. Yet, these aspects are very important and have to be brought back into prominence. 1. Probability distributions -- 2. Characteristic functions -- 3. Analytic characteristics functions -- 4. Infinitely divisible distributions -- 5. Self-decomposable distributions; triangular arrays 1. Probability distributions. 1.1. Elementary properties. 1.2. Convolutions. 1.3. Moments. 1.4. Convergence properties -- 2. Characteristic functions. 2.1. Regularity properties. 2.2. Uniqueness and inversion. 2.3. Convergence properties. 2.4. A criterion for c.f.'s. 2.5. Problems for solution -- 3. Analytic characteristic functions. 3.1. Definition and properties. 3.2. Moments. 3.3. The moment problem. 3.4. Problem for solution -- 4. Infinitely divisible distributions. 4.1. Elementary properties. 4.2. Feller measures. 4.3. Characterization of infinitely divisible distributions. 4.4. Special cases of infinitely divisible distributions. 4.5. Levy processes. 4.6. Stable distributions. 4.7. Problems for solution -- 5. Self-decomposable distributions; triangular arrays. 5.1. Self-decomposable distributions. 5.2. Triangular arrays. 5.3. Problems for solution