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

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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

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

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

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

IP, IST, José Bioucas, 2007 5 Random variables  

IP, IST, José Bioucas, 2007 6 Random variables   

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

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

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

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

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

IP, IST, José Bioucas, 2007 12 Bivariate distributions

IP, IST, José Bioucas, 2007 13 Bivariate distributions

IP, IST, José Bioucas, 2007 14 Bivariate distributions

IP, IST, José Bioucas, 2007 15 Expectation

IP, IST, José Bioucas, 2007 16 Expectation

IP, IST, José Bioucas, 2007 17 Expectation

IP, IST, José Bioucas, 2007 18 Expectation

IP, IST, José Bioucas, 2007 19 Expectation

IP, IST, José Bioucas, 2007 20 Expectation

IP, IST, José Bioucas, 2007 21 Expectation

IP, IST, José Bioucas, 2007 22 Multivariate Normal

IP, IST, José Bioucas, 2007 23 Multivariate Normal

IP, IST, José Bioucas, 2007 24 Inequalities

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

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

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