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MIT6_041F10_assn11 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem Set 11 Never Due Covered on Final Exam 1. Problem 7, page 509 in textbook Derive the ML estimator of the parameter of a Poisson random variable based of i.i.d. observa- tions X 1 , . . . , X n . Is the estimator unbiased and consistent? 2. Caleb builds a particle detector and uses it to measure radiation from far stars. On any given day, the number of particles Y that hit the detector is conditionally distributed according to a Poisson distribution conditioned on parameter x . The parameter x is unknown and is modeled as the value of a random variable X , exponentially distributed with parameter µ as follows. µx µe x 0 f X ( x ) = 0 otherwise Then, the conditional PDF of the number of particles hitting the detector is, x e x y p Y | X ( y | x ) = y ! y = 0, 1, 2, ... 0 otherwise (a) Find the MAP estimate of X from the observed particle count y . (b) Our goal is to find the conditional expectation estimator for X from the observed particle count y . i. Show that the posterior probability distribution for X given Y is of the form λ y +1 f X | Y ( x | y ) = x y e λx , x > 0 y ! and find the parameter λ . You may find the following equality useful (it is obviously true if the equation above describes a true PDF): a y +1 x y e ax dx = y ! for any a > 0 0 ii. Find the conditional expectation estimate of X from the observed particle count y .
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