Infinite entropy. This problem shows that the entropy of a discrete random variable can be infinite....

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Infinite entropy. This problem shows that the entropy of a discrete random variable can be infinite. Let A = (n log²n). [It is easy to show that A is finite by bounding the infinite sum by the integral of (x log² x)-¹.] Show that the integer-valued random vari- able X defined by Pr(X = n) = (An log² n)-1 for n = 2, 3, ..., has H(X) = +∞0. Infinite entropy. This problem shows that the entropy of a discrete random variable can be infinite. Let A = (n log²n). [It is easy to show that A is finite by bounding the infinite sum by the integral of (x log² x)-¹.] Show that the integer-valued random vari- able X defined by Pr(X = n) = (An log² n)-1 for n = 2, 3, ..., has H(X) = +∞0.

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