Question: [25] Let P be a (possibly incomputable) probability mass function, and let Pn(x) = P(x|l(x) = n) and Pn(x) = 0 for l(x) = n.

[25] Let P be a (possibly incomputable) probability mass function, and let Pn(x) = P(x|l(x) = n) and Pn(x) = 0 for l(x) = n. Show that H(Pn)−

x Pn(x)C(x) ≤ log n+ 2 log log n+O(1), where the O(1)

term is independent of P and n.

Comments. Hint: By K(x) ≤ C(x) + K(C(x)) + O(1) and the left side of Equation 8.3 on page 629 we have H(Pn) − 

 x Pn(x)C(x) ≤

x Pn(x)K(C(x)) + O(1).

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