Probability Theory Introduction

This book is designed for students who have completed proof-based courses in Advanced Calculus and Linear Algebra. In addition to covering the standard topics such as the Poisson approximation to the binomial, law of large numbers, central limit theorem, Markov chains, and simple linear regression, it also delves into several other topics and results accessible at this level, suitable for a one semester course. These include the first moment method with practical applications like cliques in the Erdos-Renyi random graph, and an upper bound on the typical longest increasing subsequence of a random permutation. The second moment method is explored with applications to Bernstein's polynomials, cliques in the Erdos-Renyi random graph, and the Hardy-Ramanujan theorem. Other topics covered include Hoeffding's inequality and the Johnson-Lindenstrauss lemma, the Hoeffding-Chernoff inequality, and the generalization ability of classification algorithms. Azuma's inequality is also discussed with various 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. The book does not assume knowledge of Lebesgue integration, although it gives some insight into why learning about it is something to look forward to through the discussion of continuous distributions. Furthermore, a number of exercises are included throughout and at the end of each section.

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