Unformatted text preview: minimum. 3. Inequality. Show ln x ≥ 11 x for x > . 4. InFnite entropy. This problem shows that the entropy of a discrete random variable can be in±nite. Let A = ∑ ∞ n =2 ( n log 2 n )1 . (It is easy to show that A is ±nite by bounding the in±nite sum by the integral of ( x log 2 x )1 .) Show that the integervalued random variable X de±ned by Pr( X = n ) = ( An log 2 n )1 for n = 2 , 3 , ... , has H ( X ) = + ∞ . 5. Markov’s 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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 '09
 Probability theory, Probability mass function, discrete random variable, Prof. T. Weissman, ndimensional probability vectors

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