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# lecture_4 - 2.160 Identification Estimation and Learning...

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2.160 Identification, Estimation, and Learning Lecture Notes No. 4 February 17, 2006 3. Random Variables and Random Processes Deterministic System: Input Output In realty, the observed output is noisy and does not fit the model perfectly. In the deterministic approach we treat such discrepancies as “error’. Random Process: An alternative approach is to explicitly model the process describing how the “error” is generated. Exogenous Input Process Observed Output Process Measurement Noise Noise Random Process Objective of modeling random processes: Use stochastic properties of the process for better estimating parameters and state of the process, and Better understand, analyze, and evaluate performance of the estimator. 3.1 Random Variables You may have already learned probability and stochastic processes in some subjects. The following are fundamentals that will be used regularly for the rest of this course. Check if you feel comfortable with each of the following definitions and terminology. (Check the box of each item below.) If not, consult standard textbooks on the subject. See the references at the end of this chapter. 1

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1) Random Variable X : Random variable is a function that maps every point in the sample space to the real axis line. F 2) Probability distribution F x (x) and probability density function f x (x) ; PDF X ( x ) = Prob( xX ) d ( f ( x ) x f ( x ) = xF ) x x dx x In the statistics and probability literature, the convention is that capital X represents a random variable while lower-case x is used for an instantiation/realization of the random variable. 3) Joint probability densities Let X and Y be two random variables f XY (x,y) = Prob (X = x and Y = y, simultaneously) 4) Statistically independent (or simply Independent) random variables f XY (x,y) = f X (x) f Y (y)
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lecture_4 - 2.160 Identification Estimation and Learning...

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