A Constructive Algebraic Proof of Student's Theorem

TL;DR

This paper presents a clear, algebraic proof of Student's theorem by explicitly constructing the orthogonal matrix involved, making the proof more accessible and suitable for educational purposes.

Contribution

It provides the first explicit algebraic construction of the orthogonal matrix used in Student's theorem proof, simplifying the understanding of the theorem.

Findings

Explicit orthogonal matrix construction for Student's theorem

Algebraic proof is complete and accessible for textbooks

Enhances understanding of the independence of sample mean and variance

Abstract

Student's theorem is an important result in statistics which states that for normal population, the sample variance is independent from the sample mean and has a chi-square distribution. The existing proofs of this theorem either overly rely on advanced tools such as moment generating functions, or fail to explicitly construct an orthogonal matrix used in the proof. This paper provides an elegant explicit construction of that matrix, making the algebraic proof complete. The constructive algebraic proof proposed here is thus very suitable for being included in textbooks.