The independence of random variable is always one tricky question. Yesterday, my friend asks me one question: X and Y1 are independent, X and Y2 are independent, so are X and (Y1, Y2) independent?
I thought that this should be right. But in fact, after one hour I could not prove it, although I have tried different ways. So I turned the mind; MAYBE IT IS WRONG!
In the next 5 minutes, I gave a very simple counter example; Let X and Y1 iid Bernoulli, but Y2 = |X - Y1|. Therefore, we can verify that P(X = 1, Y1 = 1, Y2 = 1) = 1/4, but P(X = 1) = 1/2 and P(Y1 = 1, Y2 = 1) = 1/4.
My friend tells me the prof claims this assertion? What? So I review the question in the class. Ok, the question is about the gaussian, who can be a very special case since its correlation means the dependence.
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