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Let X be a normal random variable with mean and standard deviation.pdf

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Let X be a normal random variable with mean ? and standard deviation ? and let I, independent of X, be such that P{I = 2} = P{I = -2} = 0.5. Let Y = I X. In words, Y is equally likely to be either 2X or Solution a) No, X and Y aren\'t independent Y is a linear function of X. b) Yes, I and Y are independent. I is an independent variable of both X and Y. c) Y is also a normal random variable. NORMAL DISTRIBUTION d) Cov(X,Y) = E[XY] = E[X.IX] = E[IX^2] = E[2*X^2]/2 + E[-2*X^2]/2 = (4s)/2 + (4s)/2 = 4s = 4k^2 where k is the standard deviation of X.

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Let X be a normal random variable with mean ? and standard deviation ? and let I, independent of X, be such that P{I = 2} = P{I = -2} = 0.5. Let Y = I X. In words, Y is equally likely to be either 2X or Solution a) No, X and Y aren't independent Y is a linear function of X. b) Yes, I and Y are independent. I is an independent variable of both X and Y. c) Y is also a normal random variable. NORMAL DISTRIBUTION d) Cov(X,Y) = E[XY] = E[X.IX] = E[IX^2] = E[2*X^2]/2 + E[-2*X^2]/2 = (4s)/2 + (4s)/2 = 4s = 4k^2 where k is the standard deviation of X

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