Question

Uncorrelated and Gaussian does not imply independent unless jointly Gaussian. Let X ∼N(0,1) and Y = WX, where p(W = −1) = p(W = 1) = 0 .5. It is clear that X and Y are not independent, since Y is a function of X. a. Show Y ∼N(0,1). b. Show cov[X,Y ]=0. Thus X and Y are uncorrelated but dependent, even though they are Gaussian. Hint: use the definition of covariance cov[X,Y]=E [XY] −E [X] E [Y ] and the rule of iterated expectation E [XY]=E [E [XY|W]]

Answer 1

for (b) independe of x and w is used instead of law of iterated expectation.