The purpose of this problem is to show that lack of correlation does not imply independence, even when the two random variables are Gaussian !!!
We assume that X, ∈1 and ∈2 are independent random variables, that X ∼ N(0, 1), and that P{
∈i = -1} = P{∈i = +1} = 1/2 for i = 1, 2. We define the random variable X1 and X2 by:
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1. Prove that X1 ∼ N(0, 1), X2 ∼ N(0, 1) and that ρ{X1, X2} = 0.
2. Show that X1 and X2 are not independent.