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An expectation inequality

Source: Miklós Schweitzer 2013, P12

July 12, 2014
inequalitiesratiologarithmsprobability and statsMiklos Schweitzer

Problem Statement

There are n{n} tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let Xi{X_i} be the ratio of white tokens in the pack before the ith{i^{\text{th}}} extraction and let T=max⁡{∣Xi−Xj∣:1≤i≤j≤n}. \displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}. Prove that E(T)≤H(E(X1)),{\Bbb{E}(T) \leq H(\Bbb{E}(X_1))}, where H(x)=−xln⁡x−(1−x)ln⁡(1−x).{H(x)=-x\ln x -(1-x)\ln(1-x)}.
Proposed by Tamás Móri
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