test567-10 - A-AE 567 Quiz Spring 2010 Clearly write your...

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Unformatted text preview: A-AE 567 Quiz Spring 2010 Clearly write your final answer on the exam. NAME: 1 Problem 1. Assume that x is uniform over {x : 1 ≤ |x| ≤ 2}, that is, fx (x) = γ =0 Let y = x2 . Compute Ey = if 1 ≤ |x| ≤ 2 otherwise. Find the density fy (y ) = 2 Problem 2. Assume that y = x+v where x and v are two independent random variables. The random variable v is uniform over [−1, 1] while the random variable x is uniform over {x : 1 ≤ |x| ≤ 2}, that is, fx (x) = γ =0 Let H = span{1, y}. Then compute (i) x = PH x = if 1 ≤ |x| ≤ 2 otherwise. (2) E (x − x)2 = 3 Problem 3. Consider the joint density for x and y given by fxy (x, y ) = γe−( y ) x if 0 ≤ x ≤ ∞ and 0 ≤ y ≤ 1 otherwise =0 (i) Compute the constant γ= (ii) E (x|y = y ) = g (y ) = 4 Problem 4. Consider the vectors in C3 given by 1 0 −1 φ2 = 1 x = 2 . φ1 = 1 1 −1 1 Let H = span{φ1 , φ2 }. Then compute the orthogonal projection x = PH x = αφ1 + βφ2 where α and β are constants. Recall that the inner product on C3 is given by y z 1 1 where y = y2 and z = z2 . y3 z3 (y, z ) = y1 z 1 + y2 z 2 + y3 z 3 Finally, the complex conjugate is not needed in this problem. Hint x − x is orthogonal to H. 5 Problem 5. The following questions have short answers that requires very little work: (i) Is the matrix T = positive? Answer yes or no. 2 −1 −1 2 (ii) Assume that x and y are two random variables such that E x = −3 and E (x|y) = 3 − 4y. Find PH x where H = span{1, y}. (iii) Assume that x and y are all independent random variables with mean zero and variance one. Set z = x + y. Then E z2 = (iv) Give an example of a density function whose corresponding mean is zero and variance equals one. 6 ...
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test567-10 - A-AE 567 Quiz Spring 2010 Clearly write your...

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