Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2005) Tutorial 7 Solutions Week of March 28, 2005 1. (a) a≤bfor all possible values of X, Y.Since ais the mean squared estimation error of the least squares estimator of Xbased on Y,and bis the mean squared estimation error of the linear least squares estimator of Xbased on Y,wemust have a≤b, because removing the linearity constraint can only improve the optimization, not make it worse. (b) ρX,Y= ±1 X(1−ρ2Explanation: Since b= σ2 X,Y), it is evident that the only way to pick ρX,Yto get b=0 is ρX,Y= ±1. For these two values of
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