摂南経済研究
第５巻 第 1・2 号（2015）, 1 － 5 ページ

査読付き論文Independence of the Sample Mean and Variancefor Normal Distributions: A Proof by Induction
Masayuki Hirukawa

Abstract
It is well known that the sample mean and variance of a random sample drawn from a normal
population independently follow normal and chi-squared distributions. The proof for the independence
usually relies on the condition for the independence of two quadratic forms or a linear and a quadratic
form, the orthogonal transformation, or the concept of sufficiency. This note gives a much morestraightforward proof by induction without using such advanced subjects.

Keywords: normal random sample; independence of the sample mean and variance; proof by induction.
MSC 2010 Codes: 97K70 (primary); 62H10 (secondary).

1．Introduction
Let= 1,...,

be a random sample drawn from aknown that the sample meanand variance―1―

distribution. It is well

摂南経済研究 第５巻 第１・2 号（2015）

have the following properties:
(a)
(b)
(c)

and

are independent.

Although (a) and (b) are shown straightforwardly by the transformations of normal
random variables, statistics textbooks show (c) separately by relying on the condition for
the independence of two quadratic forms or a linear and a quadratic form (e.g. Theorem
4.17 in Graybill, 1961; Theorem 2 of Chapter 12 in Hogg and Craig, 1970; and the ﬁrst
proof for Theorem 3.5.1 in Tong, 1990), the orthogonal transformation (e.g. Appendix 1.5 in
Hoel,1962; the second proof for Theorem 3.5.1 in Tong, 1990; and Theorem 6 of Chapter 9in Roussas, 1997), or the concept of sufﬁciency (e.g. Application after Theorem 9 of Chapter
11 in Roussas, 1997).

Instead, this note demonstrates (a)(b) and (c) together by induction.

The proof is free of the aforementioned subjects, which are often beyond the scope of course
work for introductory mathematical statistics.

The knowledge on moment generating

functions or characteristic functions is not presumed, either.

It may appear that the proof

requires alternative advanced subjects of differential and integral calculus including theJacobian, the Gamma and Beta functions.

However, such subjects have been applied forthe proof of (b) (the derivation of the probability density function (abbreviated as“p.d.f.”hereinafter) ofrather than (c), and thus they are not additional prerequisites.

2．The Proof
The proof starts with the case ofSolving this system for= 2.

Observe thatyields

and thus the Jacobian for the transformation from

Then, the joint p.d.f. of

becomes―2―toisIndependennce of the Sample Mean and Variance for Normal Distributions: A Proof by Induction

whereis the Gamma function.
, andandNext, suppose thatjoint p.d.f. ofare independent.
and. Consider the case of

Therefore,+1. Since

andare independent for someare mutually independent, theis(1)Now derive the joint p.d.f. ofusing (1).

ward calculations yield the transformation fromSolving this system forLet=.

Then, straightfor-toasgives(2)and thus the Jacobian for the transformation fromtois

(3)

―3―

摂南経済研究 第５巻 第１・2 号（2015）

Substituting (2) and (3) into (1) and rearranging it yield the joint p.d.f. ofas

(4)
To integrate outfrom (4) to obtain the joint p.d.f. ofLet.

, consider

Then, bythe integral becomes

Furthermore, let. Then, by,and the deﬁnition of theBeta function, this integral reduces to(5)Then, by (4) and (5), the joint p.d.f. ofﬁnally becomes―4―Independennce of the Sample Mean and Variance for Normal Distributions: A Proof by Induction

Therefore,,, andare independent.

This completes the proof by induction.

andAcknowledgments
I independently wrote up the original proof in Japanese in May 1987, when I was a juniorat the University of Tokyo.

I would like to thank Gordon Fisher for helpful comments andsuggestions on the English version.

References
Graybill, F. A. (1961): An Introduction to Linear Statistical Models , Vol.1. New York:
McGraw-Hill.
Hoel, P. G. (1962): Introduction to Mathematical Statistics , Third Edition. New York: John
Wiley & Sons.
Hogg, R. V., and A. T. Craig (1970): Introduction to Mathematical Statistics , Third Edition.
New York: Macmillan.
Roussas, G. G. (1997): A Course in Mathematical Statistics , Second Edition. San Diego:Academic Press.
Tong, Y. L. (1990): The Multivariate Normal Distribution. New York: Springer-Verlag.

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