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ProbabilityandStochasticProcesses withApplications

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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.

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

  Introduction, An introduction, Probabilityandstochasticprocesses

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