Probability Essentials

Produktbeschreibung

This introduction to Probability Theory can be used, at the beginning graduate
level, for a one-semester course on Probability Theory or for self-direction
without benefit of a formal course; the measure theory needed is developed
in the text. It will also be useful for students and teachers in related
areas such as Finance Theory (Economics), Electrical Engineering, and Operations
Research. The text covers the essentials in a directed and lean way with
28 short chapters. Assuming of readers only an undergraduate background
in mathematics, it brings them from a starting knowledge of the subject
to a knowledge of the basics of Martingale Theory. After learning Probability
Theory from this text, the interested student will be ready to continue
with the study of more advanced topics, such as Brownian Motion and Ito
Calculus, or Statistical Inference. The second edition contains some additions
to the text and to the references and some parts are completely rewritten. 

Inhaltsverzeichnis

Contents:
Introduction
Axioms of Probability
Conditional Probability and Independence
Probabilities on a Countable Space
Random Variables on a Countable Space
Construction of a Probability Measure
Construction of a Probability Measure on R
Random Variables
Integration with Respect to a Probability Measure
Independent Random Variables. Probability Distributions on R
Probability Distributions on Rn
Characteristic Functions
Properties of Characteristic Functions
Sums of Independent Random Variables
Gaussian Random Variables (The Normal and the Multivariate Normal Distributions)
Convergence of Random Variables
Weak Convergence
Weak Convergence and Characteristic Functions
The Laws of Large Numbers
The Central Limit Theorem
L2 and Hilbert Spaces
Conditional Expectation
Martingales
Supermartingales and Submartingales
Martingale Inequalities
Martingale Convergence Theorems
The Radon-Nikodym Theorem 

Kritik

"(The book is) a lean and largely self-contained introduction to the modern theory of probability, aimed at advanced undergraduate or beginning graduate students. The 28 short chapters belie the book's genesis as polished lecture notes; the exposition is sleek and rigorous and each chapter ends with a supporting collection of mainly routine exercises. ... The authors make it clear what luggage is required for this exhilarating trek,... a good knowledge of advanced calculus, some linear algebra, and some "mathematical sophistication". With this understood, the itinerary is immaculately paced and planned with just the right balances of technical ascents and pauses to admire the scenery. Within the constraints of a slim volume, it is hard to imagine how the authors could have done a more effective or more attractive job." The Mathematical Gazette, Vol. 84, No 500, 2000 "The authors provide the shortest path through the twenty-eight chapter headings. The topics are treated in a mathematically and pedagogically digestible way. The writing is concise and crisp: the average chapter length is about eight pages. ... Numerous exercises add to the value of the text as a teaching tool. In conclusion, this is an excellent text for the intended audience."
Short Book Reviews, Vol. 21, No. 2, 2001