ps08 - Massachusetts Institute of Technology Department of...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Problem Set 8 Topics: Covariance, Estimation, Limit Theorems Due: April 26, 2006 1. Consider n independent tosses of a k -sided fair die. Let X i be the number of tosses that result in i . Show that X 1 and X 2 are negatively correlated (i.e., a large number of ones suggests a smaller number of twos). 2. Oscar’s dog has, yet again, run away from him. But, this time, Oscar will be using modern technology to aid him in his search: Oscar uses his pocket GPS device to help him pinpoint the distance between him and his dog, X miles. The reported distance has a noise component, and since Oscar bought a cheap GPS device the noise is quite significant. The measurement that Oscar reads on his display is random variable Y = X + W (in miles) , where W is independent of X and has the uniform distribution on [ - 1 , 1]. Having knowledge of the distribution of X lets Oscar do better than just use Y as his guess of the distance to the dog. Oscar somehow knows that X is a random variable with the uniform distribution on [5 , 10]. (a) Determine an estimator g ( Y ) of X that minimizes E [( X - g ( Y )) 2 ] for all possible mea- surement values Y = y . Provide a plot of this optimal estimator as a function of y . (b) Determine the linear least squares estimator of X based on Y . Plot this estimator and compare it with the estimator from part (a). (For comparison, just plot the two estimators on the same graph and make some comments.) 3. (a) Given the information E [ X ] = 7 and var( X ) = 9, use the Chebyshev inequality to find a lower bound for P (4 X 10).
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