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mid1(07)sol

# mid1(07)sol - nctuee07f Stochastic Processes Midterm 1...

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nctuee07f Stochastic Processes Midterm 1 Solutions 1. True or False (6 points × 5 = 30 points) Please provide an explanation, a simple proof or a counter example, to your an- swer. No points will be credited if answers are with true or false only . Try to be concise and to the point. (a) True . This is just the singular value decomposition applied to a square matrix. (b) True . For Hermitian matrix A , we know A = A H . The eigenvector x associated with the eigenvalue λ satisfies Ax = λ x . It follows that λ · x H x = x H · λ x = x H Ax = x H A H x since A = A H = Ax H x = λ * · x H x , from which we know λ = λ * . Thus, the eigenvalue of a Hermitian matrix is always real. (c) False . We need to add that X and Y are jointly Gaussian in order to make such conclusion. (d) True . See HW1 solutions. (e) True . Consider independent discrete random variables X and Y . By independence we know p X,Y ( x, y ) = p X ( x ) p Y ( y ) . Then, E [ XY ] = X x X y xyp X,Y ( x, y ) = X x X y xyp X ( x ) p Y ( y ) = X x xp X ( x ) · X y yp Y ( y ) = E [ X ] E [ Y ] . Thus, X and Y are uncorrelated. 1

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2. Central Limit Theorem (10 points) Let Y = N X i =1 X i .
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mid1(07)sol - nctuee07f Stochastic Processes Midterm 1...

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