Show that lack of correlation does not imply independence

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Reference no: EM131231684

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 X₂are not independent.

Reference no: EM131231684

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