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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 zeromean 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 .
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b Express the covariance covX; Y between X and Y in terms of a; b; c; X .
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c Are X and Y independent? Why?
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d Can X and Y be uncorrelated? If so, when are X and Y be uncorrelated? If not,
explain why.
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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 ?
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b Write down the joint p.d.f. fXY x; y between X and Y .
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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.
 Fall '08
 Staff

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