11SFinal

11SFinal - Final Exam of ECE302 Prof Wang’s section...

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Unformatted text preview: Final Exam of ECE302, Prof. Wang’s section 3:20–5:20pm Thursday, May 5, 2011, CL50 224. 1. Please make sure that it is your name printed on the exam booklet. Enter your student ID number, e-mail address, and signature in the space provided on this page, NOW! 2. This is a closed book exam. 3. You have two hours to complete it. The students are suggested not spending too much time on a single question, and working on those that you know how to solve. 4. There are 16 pages in the exam booklet. Use the back of each page for rough work. 5. Neither calculators nor help sheets are allowed. Name: Student ID: E-mail: Signature: Question 1: [20%, Work-out question] Consider a random variable X . The probabilities X = 1 and X =- 1 are 1/2 and 1/2, respectively. We are not able to directly observe X . Instead, we observe Y = X + N where N is a random variable with probability density function f N ( n ) = 0 . 5 e-| n | . (1) X and N are independent. 1. [10%] Find out the linear MMSE estimator of X given Y = y . That is, your answer should be a function ˆ X lin.MMSE ( y ). Note: If you do not know how to solve this problem, write down what MMSE stands for and you will receive 4 points. 2. [10%] Find out the MMSE estimator of X given Y = y . That is, your answer should be a function ˆ X MMSE ( y ). Note: If you do not know how to solve this problem, answer which estimator has better performance: a MMSE estimator or a linear MMSE estimator. You will receive 2 points. Question 2: [10%, Work-out question] X 1 , X 2 , and X 3 are three independent standard Gaussian random variable. Construct three new random variables by Y 1 = X 1 + X 2 + X 3 (2) Y 2 = 2 X 1- X 2- X 3 (3) Y 3 = X 2- X 3 (4) 1. [4%] Find the mean vector and the covariance matrix of ( Y 1 ,Y 2 ,Y 3 )....
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This note was uploaded on 02/12/2012 for the course ECE 302 taught by Professor Gelfand during the Spring '08 term at Purdue.

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11SFinal - Final Exam of ECE302 Prof Wang’s section...

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