An example of dependent variables with zero correlation coefficient. We consider the
following two cases. Let X be uniformly distributed
(I) on the set {0,1,2, . . . ,N};
(II) on the set {−N, − N + 1, . . . , − 1,0,1, . . . ,N}.
Define another, clearly dependent, variable Y simply by assuming Y = X².
(a) Show that the correlation coefficient between X and Y is non-zero for case (I) and zero for case (II).
(b) Show that the mutual information between X and Y is non-zero both for case (I) and case (II).