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ps4 - Massachusetts Institute of Technology Department of...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.432 Stochastic Processes, Detection and Estimation Problem Set 4 Spring 2004 Issued: Thursday, February 26, 2004 Due: Thursday, March 4, 2004 Reading: This problem set: Sections 1.7, 3.2.5, 3.3.1, 3.3.2 Next: Sections 3.3.3 - 3.3.6, 3.4 Exam #1 Reminder: Our first quiz will take place Thursday, March 11, 9 am - 11 am. The exam will cover material through Lecture 8 (March 6), as well as the associated Problem Sets 1 through 4. You are allowed to bring one 8 1 �� × 11 �� sheet of notes (both sides). Note that there 2 will be no lecture on March 11. Problem 4.1 Suppose w , z are scalar random variables, and that 1 / 2 | z | < 1 p z ( z ) = . 0 otherwise You are told that the Bayes least-squares estimate of w given an observation z is 1 1 / 2 z 0 ˆ w BLS = sgn z = , 2 1 / 2 otherwise and the associated mean-square estimation error is BLS = 1 / 12. However, you would prefer to use the following ad-hoc estimator ˆ w AH = z . w AH ) = E [ w ˆ (a) Is it possible to determine b ( ˆ w AH ], the bias of your new esti- mator, from the information given? If your answer is no, briefly explain your reasoning. If your answer is yes, calculate b ( ˆ w AH ). (b) Is it possible to determine AH = E [( w ˆ w AH ) 2 ], the mean-square estimation error obtained using this estimator, from the information given? If your
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