Show that lack of correlation does not imply independence

Question

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:

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.