probability

probability - Notes on Probability A.E. Frazho 2 Contents 1...

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Unformatted text preview: Notes on Probability A.E. Frazho 2 Contents 1 The Probability Measure 7 1.1 The axioms of probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 The birthday problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 The gamblers ruin problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Bayes Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.1 Drug testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.2 A classical example of Bayes rule. . . . . . . . . . . . . . . . . . . . . 17 1.5.3 The Monty Hall Problem. . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Independent events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.7 PageRank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.7.1 Markov matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Random Variables 35 2.1 The distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 The density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3 The exponential density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 The uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 The Gaussian random variable . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.1 An ane function of a random variable . . . . . . . . . . . . . . . . . 46 2.6 The diusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7 The binomial random variable . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.8 The geometric random variable . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.9 The Poisson random variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.10 Functions of a uniform random variable . . . . . . . . . . . . . . . . . . . . . 54 2.11 The joint distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.11.1 Independent random variables. . . . . . . . . . . . . . . . . . . . . . 58 2.12 The joint density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.12.1 Independent random variables. . . . . . . . . . . . . . . . . . . . . . 61 2.13 A uniformly distributed dart board. . . . . . . . . . . . . . . . . . . . . . . . 62 2.14 The Raleigh density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.14.1 The Box-Muller transform . . . . . . . . . . . . . . . . . . . . . . . . 66 2.15 The maximum and minimum of random variables . . . . . . . . . . . . . . . 69 2.15.1 The minimum of random variables . . . . . . . . . . . . . . . . . . . 70 3 4 CONTENTS 2.16 The Buon needle problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.17 The meeting problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3 Expectation...
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probability - Notes on Probability A.E. Frazho 2 Contents 1...

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