MathDB

12

Part of 2013 Miklós Schweitzer

Problems(1)

An expectation inequality

Source: Miklós Schweitzer 2013, P12

7/12/2014
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
inequalitiesratiologarithmsprobability and statsMiklos Schweitzer