Introduction to Probability Theory - Softcover

Panchenko, Dmitry

Softcover

ISBN 10: 1999419022 ISBN 13: 9781999419028

Publisher: Dmitriy Panchenko, 2019

Synopsis

This book is intended for students who have completed proof-based courses in Advanced Calculus and Linear Algebra. In addition to the standard topics (such as the Poisson approximation to the binomial, law of large numbers, central limit theorem, Markov chains, and simple linear regression), several other topics and results that are accessible at this level and that fit into a one semester course are covered:--first moment method with some applications, such as cliques in the Erdos-Renyi random graph, and an upper bound on the typical longest increasing subsequence of a random permutation; --second moment method with applications to Bernstein's polynomials, cliques in the Erdos-Renyi random graph, and the Hardy-Ramanujan theorem; --Hoeffding's inequality, and the Johnson-Lindenstrauss lemma; --the Hoeffding-Chernoff inequality, and the generalization ability of classification algorithms; --Azuma's inequality with several examples, such as the chromatic number of the Erdos-Renyi random graph, max-cut in sparse random graphs, and the Hamming distance on the hypercube. A knowledge of Lebesgue integration is not assumed, although the discussion of continuous distributions gives some idea of why learning about it is something to look forward to. A number of exercises are included throughout and at the end of each section. In the second edition, a number of explanations were clarified and some mistakes were corrected.

About this title

About the Author

Dmitry Panchenko is a Professor of Mathematics at the University of Toronto. He works in Probability Theory, with most of his work focused on spin glass models and concentration inequalities with application to statistical learning theory.

From the Inside Flap

Introduction

• Example: Balancing Vectors
• Discrete Probability Spaces and Distributions
• Poisson Approximation to the Binomial
• Independence. Conditional Distributions
• Applications of the Linearity of Expectation

Second Moment Calculations

• Variance and Covariance
• Classical Law of Large Numbers
• Bernstein Polynomials
• Cliques in the Erdos-Renyi Random Graph
• Hardy-Ramanujan Theorem

Exponential Inequalities

• Hoeffding Inequality
• Johnson-Lindenstrauss Lemma
• Hoeffding-Chernoff Inequality
• Azuma Inequality
• Applications of Azuma Inequality

Gaussian Distributions

• Gaussian Distributions on R
• Gaussian Distributions on R^n
• Central Limit Theorem
• Distributions Related to Gaussian
• Simple Linear Regression

Finite State Markov Chains

• Definitions and Basic Properties
• Stationary Distributions
• Convergence Theorem
• Reversible Markov Chains

Edition details

• Publisher: Dmitriy Panchenko
• Publication date: 2019
• Language: English
• ISBN 10: 1999419022
• ISBN 13: 9781999419028
• Binding: Paperback
• Number of pages: 250