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Maximal expectation of distance on 2-sphere

Source: Miklós Schweitzer 2016, Problem 10

November 2, 2016

geometric probabilitysphereprobability distributionprobability and statsreal analysiscollege contestsMiklos Schweitzer

Problem Statement

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?