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Unformatted text preview: EE 562a Midterm Exam A SOLUTIONS, 28 Februrary 2007 WRITE YOUR NAME on the exam paper, and indicate if you are an ONCAMPUS or DEN student. You may use a simple calculator (i.e., not a scientific calculator), and one lettersized sheet of notes (both sides). Do all problems. Note that these problems have different point values, and differ widely in difficulty. You should look over all the problems before starting your work. Write all answers on the test paper. You may use scratch paper to do the problem, but the answer must be written with the problem on the exam paper. If you do work on scratch paper, and wish that work to be considered for partial credit, hand in the scratch paper along with the exam. Part I. Basic Concepts. For each question in this section, circle the best answer from the choices listed. Each problem has only one best answer. There is no partial credit on this section. (3 points each.) 1. Under which of the following conditions are the random variables x ( u ) and y ( u ) both orthog onal and uncorrelated? (a) E { x ( u ) } = E { x ( u ) y ( u ) } = 0 (b) E { x ( u ) y ( u ) } = δ ( x y ) (c) E { x ( u ) y ( u ) } = E { x ( u ) } E { y ( u ) } (d) E { x ( u ) y ( u ) } = 0 2. Which of the following is an affine transformation? (a) ( x, y ) → xy (b) x →  x  (c) x → b for a constant nvector b (d) x → ( x 2 , x 3 ) T 3. Which of the following could be a covariance matrix K xx * ? (a) 2 2 1 2 2 (b) 3 3 i 4 3 i 5 (c) 2 2 1 2 2 (d) 3 2 1 2 1 3 2 3 1 4. Suppose that a circular random dvector x ( u ) has a d × d covariance matrix with rank r < d . Which of the following is possible by an affine transformation? (a) x ( u )→ ddimensional white noise (b) x ( u )→ kdimensional white noise, k < r (c) x ( u )→ y ( u ) with K yy 6 = (d) x ( u )→ y ( u ) with K yy * a d × d matrix with determinant 1 EE 562a Midterm Exam SOLUTIONS 2 5. Suppose that a random vector x ( u ) has a correlation matrix R x = 2 1 1 2 . Which of the following could be its mean vector m x ? (a) p 1 / 2 p 1 / 2 (b) 2 2 (c) 1 1 (d) Any vector m x is possible 6. Suppose x ( u ) and y ( u ) are jointly Gaussian real random vectors. Which of the following gives enough information to find their joint probability density function? (a) K x and K y (b) R x , R y , m x and m y (c) R x , K y , m y , and that x ( u ) is meanzero (d) R x , R y , m x , m y , and that x ( u ) and y ( u ) are uncorrelated 7. Which of the following could be allowed as a preprocessing step without loss of information (with no restriction on the type of allowed operations)?(with no restriction on the type of allowed operations)?...
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 Spring '07
 ToddBrun
 Variance, Probability theory, Midterm Exam Solutions, KXX

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