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A wicked game of expectations... and integral probability functions

Source: MIPT Undergraduate Competition 2019 2.2

August 17, 2020
inequalitiesprobability and statsexpectationcalculusintegrationprobabilityfunction

Problem Statement

Petya and Vasya are playing the following game. Petya chooses a non-negative random value ξ\xi with expectation E[ξ]=1\mathbb{E} [\xi ] = 1, after which Vasya chooses his own value η\eta with expectation E[η]=1\mathbb{E} [\eta ] = 1 without reference to the value of ξ\xi. For which maximal value pp can Petya choose a value ξ\xi in such a way that for any choice of Vasya's η\eta, the inequality P[η≥ξ]≤p\mathbb{P}[\eta \geq \xi ] \leq p holds?
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