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2016 Miklós Schweitzer, 10

Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance of $X$ and $Y$ be maximal?

2016 Miklós Schweitzer, 9

Tags: probability and stats , probability distribution , probability

For $p_0,\dots,p_d\in\mathbb{R}^d$, let \[ S(p_0,\dots,p_d)=\left\{ \alpha_0p_0+\dots+\alpha_dp_d : \alpha_i\le 1, \sum_{i=0}^d \alpha_i =1 \right\}. \] Let $\pi$ be an arbitrary probability distribution on $\mathbb{R}^d$, and choose $p_0,\dots,p_d$ independently with distribution $\pi$. Prove that the expectation of $\pi(S(p_0,\dots,p_d))$ is at least $1/(d+2)$.

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