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ProbabilityandStochasticProcesses withApplications

Probability and Stochastic Processeswith ApplicationsOliver KnillContentsPreface31 What is probability theory? .. Some paradoxes in probability theory .. Some applications of probability theory .. 182 Limit Probability spaces, random variables, independence .. Kolmogorov s 0 1 law, Borel-Cantelli lemma .. Integration, Expectation, Variance .. Results from real analysis .. Some inequalities .. The weak law of large numbers .. The probability distribution function .. Convergence of random variables .. The strong law of large numbers .. The Birkhoff ergodic theorem .. More convergence results .. Classes of random variables .. Weak convergence .. The central limit theorem .. Entropy of distributions .. Markov operators .. Characteristic functions .. The law of the iterated logarithm .. 1233 Discrete Stochastic Conditional Expectation.

Preface These notes grew from an introduction to probability theory taught during the first and second term of 1994 at Caltech. There was a mixed audience of

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