# hw2 - Maximize h Z V x V y V z Z ≥ V x V y V z ∈ R 3...

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EE 376B/Stat 376B Handout #6 Information Theory Thursday, April 13, 2006 Prof. T. Cover Due Thursday, April 20, 2006 Homework Set #2 1. Maximum entropy with marginals. What is the maximum entropy probability mass function p ( x, y ) with the following marginals? You may wish to guess and verify a more general result. y 1 y 2 y 3 x 1 p 11 p 12 p 13 1 / 2 x 2 p 21 p 22 p 23 1 / 4 x 3 p 31 p 32 p 33 1 / 4 2 / 3 1 / 6 1 / 6 2. Processes With Fixed Marginals Consider the set of all densities with fixed pairwise marginals f 12 ( x 1 , x 2 ) , f 23 ( x 2 , x 3 ) , . . . , f n - 1 ,n ( x n - 1 , x n ) . Show that the maximum entropy process with these marginals is the first-order (pos- sibly time-varying) Markov process with these marginals. Identify the maximizing f * ( x 1 , x 2 , . . . , x n ). 1

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3. Maximum entropy of atmosphere.
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Unformatted text preview: Maximize h ( Z, V x , V y , V z ) , Z ≥ , ( V x , V y , V z ) ∈ R 3 , subject to the energy constraint E ( 1 2 m k V k 2 + mgZ ) = E . Show that the resulting distribution yields E 1 2 m k V k 2 = 3 5 E EmgZ = 2 5 E . Thus 2 5 of the energy is stored in the potential Feld, regardless of its strength g . 4. Maximum entropy processes. ±ind the maximum entropy rate stochastic processes { X i } ∞-∞ subject to the constraints: (a) EX 2 i = 1 , i = 1 , 2 , . . . , (b) EX 2 i = 1, EX i X i +1 = 1 2 , i = 1 , 2 , . . . . ±ind the maximum entropy spectrum for the processes in parts (a) and (b). 2...
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