1 02 1 03 1 04 0 x 1 2 suppose that y ax bx

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Unformatted text preview: 1 = 0:4 : y 0.1 0.2 1 0.3 -1 0.4 0 x 1 2. Suppose that Y = aX + bX + c, and X is a Gaussian r.v. with zero-mean and variance X 6= 0; i.e., X  N 0; X . a Express the mean E Y of Y in terms of a; b; c; X . 4 b Express the covariance covX; Y  between X and Y in terms of a; b; c; X . 6 c Are X and Y independent? Why? 2 d Can X and Y be uncorrelated? If so, when are X and Y be uncorrelated? If not, explain why. 3 2 2 2 2 2 Solution: a mY = E Y = a b 2 X +c covX; Y  = E X Y , mX mY = E X Y = E aX + bX + cX = b Note: E X = 0 because X is symmetrically distributed. 3 3 2 2 2 X c Since Y is a function of X , they are dependent . d The covariance between X and Y is zero when b = 0 , in which case X and Y are uncorrelated. 3. Let X; Y; Z be statistically independent r.v.'s, uniformly distributed between ,3 and 3. a What is E X Y + Z ? 10 b Write down the joint p.d.f. fXY x; y between X and Y . 5 2 2 Solution: a First, we nd E X = E Y = E Z = 0 because the random variables are symmetric...
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This note was uploaded on 03/10/2013 for the course ECE 3075 taught by Professor Staff during the Fall '08 term at Georgia Institute of Technology.

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