tut08 - Find the numerical value o E [ g ( X )] and var( g...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Tutorial 0 8 April 13-14, 2006 1. Suppose X is a unit normal random variable. Defne a new random variable Y such that: Y = a + bX + cX 2 . Find the correlation coefficient ρ ±or X, Y . 2. Continuous random variables X and Y have a joint PDF given by C i± ( x, y ) belongs to the closed shaded region f X,Y ( x, y ) = 0 otherwise y x 2 1 1 2 (a) The experimental value o± X will be revealed to us; we have to design an estimator g ( X ) Y that minimizes the conditional expectation E [( Y g ( X )) 2 | X = x ], ±or all x , over all possible estimators. Provide a plot o± the optimal estimator as a ±unction o± its argument. (b) Let g ( X ) be the optimal estimator o± part (a).
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Unformatted text preview: Find the numerical value o E [ g ( X )] and var( g ( X ))? (c) Find the mean square error E [( Y g ( X )) 2 ]. Is that the same as E [var( Y | X )]? (d) Find var( Y ). 3. Random variable X is uniormly distributed between -1.0 and 1.0. Let X 1 , X 2 , . . . , be independent identically distributed random variables with the same distribution as X . Determine which, i any, o the ollowing sequences (all with i = 1 , 2 , . . . ) are convergent in probability. Give reasons or your answers. Include the limits i they exist. (a) X i X i (b) Y i = i (c) Z i = ( X i ) i (d) T i = X 1 + X 2 + . . . + X i X 1 + X 2 + . . . + X i (e) U i = i () V i = X 1 X 2 . . . X i (g) W i = max( X 1 , . . . , X i ) Page 1 o 2...
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