problem set3 - ELEC 531 STATISTICAL SIGNAL PROCESSING...

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ELEC 531: STATISTICAL SIGNAL PROCESSING Department of Electrical and Computer Engineering Rice University Fall 2007 Problem Set 3 Due: September 20, 2007 Problem 3.1 We make L observations X 1 , . . . , X L of a parameter θ corrupted by additive noise ( X l = θ + N l ). The parameter θ is a Gaussian random variable [ θ ∼ N ( m θ , σ 2 θ )] and N l are statistically independent Gaussian random variables [ N l ∼ N (0 , σ 2 N )]. (a) Find b θ MMSE ( X ), the MMSE estimator of θ . (b) Assuming m θ = 0, compute the resulting mean-squared error. (c) Find the conditional mean-squared error of b θ MMSE ( X ), i.e. compute E X | θ £ ( θ - b θ MMSE ) 2 | θ / . How does this answer compare with the result in part (b)? Again assume that m θ = 0. (d) Find the maximum a posteriori ( MAP ) estimator of θ defined as the maximizer of the a posteriori density, b θ MAP = arg max θ p ( θ | X ) . Compare this estimator to the MMSE estimator found in part (a). Assume m θ has a non-zero value. Problem 3.2 Imperfect Geiger Counter. A radioactive source emits N radioactive parti-
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