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**Unformatted text preview: **minimum. 3. Inequality. Show ln x 1-1 x for x > . 4. InFnite entropy. This problem shows that the entropy of a discrete random variable can be innite. Let A = n =2 ( n log 2 n )-1 . (It is easy to show that A is nite by bounding the innite sum by the integral of ( x log 2 x )-1 .) Show that the integer-valued random variable X dened by Pr( X = n ) = ( An log 2 n )-1 for n = 2 , 3 , ... , has H ( X ) = + . 5. Markovs inequality or probabilities. Let p ( x ) be a probability mass function. Prove, for all d 0, Pr { p ( X ) d } log p 1 d P H ( X ) . (1) 1...

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