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Let $x_1,x_2,\dots,x_n$ by random variables which take the values $0$ or $1$ with $P(x_i = 1) = p_i$ and $P(x_i = 0) = 1-p_i$, where $0 \leq p_i \leq 1$ for $i=1,2,\dots, n$. Let $$X= \sum_{i=1}^n x_i.$$

Is it true that the entropy of $X$ is maximal when $p_i = 1/2$ for $i=1,2,\dots, n$?

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True but not very easy statweb.stanford.edu/~ckirby/techreports/NSF/… –  leonbloy Jan 6 at 17:32
    
@leonbloy Thanks for the reference. I will leave the question open in case anyone has a nice way of explaining it an answer here. –  user115998 Jan 6 at 21:05

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