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midterm2_S08

# midterm2_S08 - six numbers De±ne two random variables X...

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April 4, 2008 Spring 2008 Optimal Estimation (EGM 6934, Sec 4159 ) University of Florida Mechanical and Aerospace Engineering Instructor: Prabir Barooah Midterm 2 Due: in class on Monday, April 7, 2008 Points will be awarded for clarity and completeness of your answers. Write your name on all of your answer sheets before turning them in. Problem 1. [5 pt] Explain whether the following is true or not: E [ 1 X ] = 1 E [ X ] ? Problem 2. [5 pt] Let Z = + ǫ , where θ is a deterministic parameter vector, ǫ is zero mean, and R = Cov ( ǫ, ǫ ). However, H is random. Consider the following proposed estimator: ˆ θ = ( H T R - 1 H ) - 1 H T R - 1 Z When is ˆ θ an unbiased estimator of θ ? Problem 3. [10 pt] Let { ω 1 , . . . , ω 6 } be the outcomes of casting a die (i.e., ω i = number of dots that appear). Assume that P ( { ω i } ) = 1 6 for i = 1 , . . . , 6, i.e., the probability of getting a number is 1 6 for all the six numbers. Define two random variables
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Unformatted text preview: six numbers. De±ne two random variables X and Y as: X ( ω ) = b 1 ω is odd ω is even Y ( ω ) = b ω is odd 1 ω is even Are X and Y independent? Provide clear arguments to support your answer. Problem 4. [15 pt] Let X be a random variable that has a so-called Laplacian distribution, i.e., its pdf is given by f X ( x ) = 1 2 λ e-1 λ | x | ,-∞ < x < ∞ where λ is a deterministic parameter 1 . Let X 1 ,...,X N be independent random variables with the same exponential distribution as X . You can imagine these as observations of X that you have not collected yet. Design an unbiased estimator of λ that uses X 1 ,...,X N . 1 This distribution is important in image and speech analysis Prabir Barooah 1...
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• Fall '08
• Staff
• Probability theory, unbiased estimator, Prabir Barooah, Aerospace Engineering Instructor, following proposed estimator

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