In Section 1.6, we introduced the idea of entropy h(x) as th
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In Section 1.6, we introduced the idea of entropy h(x) as the information gained on observing the value of a random vari
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In Section 1.6, we introduced the idea of entropy h(x) as the information gained on observing the value of a random variable x having distribution p(x). We saw that, for independent variables x and y for which p(x, y) = p(x)p(y), the entropy functions are additive, so that h(x, y) = h(x) + h(y). In this exercise, we derive the relation between h and p in the form of a function h(p). First show that h(p2) = 2h(p), and hence by induction that h(pn) = nh(p) where n is a positive integer. Hence show that h(pn/m)=(n/m)h(p) where m is also a positive integer. This implies that h(px) = xh(p) where x is a positive rational number, and hence by continuity when it is a positive real number. Finally, show that this implies h(p) must take the form h(p) ? ln p.
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