Suppose that we four random variables, A, B, C, D that have the same standard deviation. Here, A, B, C, and D are chosen randomly and independently of each other. Define E = -A-B+kC+(2-k)D and F = -A-B+(2-k)C+kD, where k is a real constant. Find the set of values of k such that the variables E and F have zero correlation.
Upon experimenting with large randomly-generated data sets of Microsoft Excel, I am led to believe that the possible values of k are 1 +- sqrt(2), but I have no idea how to prove that these are the possible values of k.
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