Youden's Demon is Sylvester's Problem

TL;DR

This paper reveals a surprising equivalence between a probability problem involving Gaussian points and a geometric problem about convex positions, generalizing Sylvester's Four Point problem to higher dimensions.

Contribution

It establishes a connection between Gaussian order statistics and Sylvester's problem, extending the classical problem to Gaussian points in multiple dimensions.

Findings

Probability of two above-average heights equals convex position probability (~0.649).

Gale duality links Gaussian points with conditioned independent samples.

Generalizes Sylvester's Four Point problem to Gaussian points in .

Abstract

If four people with Gaussian-distributed heights stand at Gaussian positions on the plane, the probability that there are exactly two people whose height is above the average of the four is exactly the same as the probability that they stand in convex position; both probabilities are $\frac 6 \pi \arcsin\left(\frac{1}{3}\right)\approx .649$. We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line ("Youden's demon problem") is the same as a natural generalization of Sylvester's Four Point problem to Gaussian points in $\mathbb{R}^d$. Our main tool is the observation that the Gale dual of independent samples in $\mathbb{R}^d$ itself can be taken to be a set of independent points (conditioned on barycenter at the origin) when the distribution of the points is Gaussian.