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IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty Observation mechanism: Priors: Parameters Role in inverse problems

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IP, IST, José Bioucas, 2007 2 Overview of Probability Definition; Properties Independency; Conditional probability; Bayes theorem Random variables; Cumulative distribution function Examples of random variables Bivariate distributions; Marginal distribution;Conditional distribution Multivariate distributions; Marginal distribution;Conditional distributions Expectation of a random variable; Variance; Covariance Conditional expectation of a random variable; Variance; Covariance Weak law of large numbers Ref. Larry Wasserman, All of Statistics. A Concise Course in Statistical Inference, Springer, 2004

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IP, IST, José Bioucas, 2007 3 Definition Frequencist interpretation: Number of occurencies of A Number of repetitions Bayesain interpretation: Measures an observer’s strength of belief that A is true In probability the interpretation does not matter sample space event

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IP, IST, José Bioucas, 2007 4 Independency; Conditional probability

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IP, IST, José Bioucas, 2007 5 Random variables

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IP, IST, José Bioucas, 2007 6 Random variables

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IP, IST, José Bioucas, 2007 7 Some discrete random variables From Wikipedia

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IP, IST, José Bioucas, 2007 8 Some discrete random variables From Wikipedia

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IP, IST, José Bioucas, 2007 9 Some continuous random variables

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IP, IST, José Bioucas, 2007 10 Some continuous random variables

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IP, IST, José Bioucas, 2007 11 Some continuous random variables

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IP, IST, José Bioucas, 2007 12 Bivariate distributions

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IP, IST, José Bioucas, 2007 13 Bivariate distributions

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IP, IST, José Bioucas, 2007 14 Bivariate distributions

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IP, IST, José Bioucas, 2007 15 Expectation

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IP, IST, José Bioucas, 2007 16 Expectation

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IP, IST, José Bioucas, 2007 17 Expectation

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IP, IST, José Bioucas, 2007 18 Expectation

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IP, IST, José Bioucas, 2007 19 Expectation

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IP, IST, José Bioucas, 2007 20 Expectation

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IP, IST, José Bioucas, 2007 21 Expectation

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IP, IST, José Bioucas, 2007 22 Multivariate Normal

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IP, IST, José Bioucas, 2007 23 Multivariate Normal

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IP, IST, José Bioucas, 2007 24 Inequalities

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IP, IST, José Bioucas, 2007 25 Laws of large numbers

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IP, IST, José Bioucas, 2007 26 Central Limit Theorem

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