Closed forms and open obstructions: the sample variance of three observations
Abstract
The exact distribution of the sample variance for n=3 is known since Rietz (1931) for the uniform parent, and Royen (2007, 2008) gave a Fourier series for any bounded continuous parent. Neither settles which parents admit a finite closed form, nor which special functions it forces. In coordinates aligned with the cube diagonal the variance constraint becomes a cylinder and the cube cross-section a polygon with S3 symmetry, hexagonal over the central band of the diagonal and triangular near either corner; the CDF is the volume of their intersection, the parent density entering as a weight. A closure hierarchy is then organized by the minimal function class containing the parent density: polynomial parents on any bounded interval always close in elementary terms, a theorem for the whole class; in the negative direction a single explicit parent already suffices, and for a rational one we prove the CDF is not elementary, the obstruction being an irreducible dilogarithmic part. Beyond those two theorems the hierarchy is a set of example-specific obstructions rather than a classification: for an algebraic parent the radial first-kind differential is shown non-elementary on a genus-two curve, and the exponential row is a conjecture supported by the Bessel structure of its radial integral. For the uniform parent we obtain a two-piece formula bifurcating at Y=1/4, where the variance disk first circumscribes the hexagonal cross-section at the cube center. For the singular arcsine parent we derive both endpoint laws in closed form and give a six-term approximation accurate to about 10−3, an accuracy Royen's universal series reaches at about a hundred terms.
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BibTeX
@misc{osan2026closed,
title = {{Closed forms and open obstructions: the sample variance of three observations}},
author = {Remus Osan and Kevin T. Chu and Ron Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25333},
eprint = {2609.25333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25333},
}
RIS
TY - UNPB
TI - Closed forms and open obstructions: the sample variance of three observations
AU - Remus Osan
AU - Kevin T. Chu
AU - Ron Yu
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25333
UR - https://arxiv.org/abs/2609.25333
AB - The exact distribution of the sample variance for n=3 is known since Rietz (1931) for the uniform parent, and Royen (2007, 2008) gave a Fourier series for any bounded continuous parent. Neither settles which parents admit a finite closed form, nor which special functions it forces. In coordinates aligned with the cube diagonal the variance constraint becomes a cylinder and the cube cross-section a polygon with S_3 symmetry, hexagonal over the central band of the diagonal and triangular near either corner; the CDF is the volume of their intersection, the parent density entering as a weight. A closure hierarchy is then organized by the minimal function class containing the parent density: polynomial parents on any bounded interval always close in elementary terms, a theorem for the whole class; in the negative direction a single explicit parent already suffices, and for a rational one we prove the CDF is not elementary, the obstruction being an irreducible dilogarithmic part. Beyond those two theorems the hierarchy is a set of example-specific obstructions rather than a classification: for an algebraic parent the radial first-kind differential is shown non-elementary on a genus-two curve, and the exponential row is a conjecture supported by the Bessel structure of its radial integral. For the uniform parent we obtain a two-piece formula bifurcating at Y=1/4, where the variance disk first circumscribes the hexagonal cross-section at the cube center. For the singular arcsine parent we derive both endpoint laws in closed form and give a six-term approximation accurate to about 10^−3, an accuracy Royen's universal series reaches at about a hundred terms.
ER -