Let X1,X2 and X3 be three discrete random variables with
P[X1 = 0] = P[X1 = 1] = P[X2 = 0] = P[X2 = 1] = 1/2
and P[X3 = 0] = 1.
(i) Characterize all possible coupling between X1 and X2.
(ii) Which coupling maximizes the correlation? Which coupling minimizes the
correlation? Do you have an intuitive explanation why these couplings are the
ones that minimize/maximize the correlation?
(iii) Which coupling makes the two random variables uncorrelated?
(iv) Do the tasks (i) − (iii) but for X1 and X3
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Let X1,X2 and X3 be three discrete random variables with P[X1 = 0] = P[X1 = 1] = P[X2 = 0] = P[X2 = 1] = 1 2 and P[X3 = 0] = 1. (i) Characterize all possible coupling between X1 and X2. (ii) Which coupling maximizes the correlation? Which coupling minimiz
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