Three Remarkable Properties of the Normal Distribution

12 Pages Posted: 27 Oct 2018

See all articles by Eric Benhamou

Eric Benhamou

A.I. Square Connect; LAMSADE- Paris Dauphine University

Beatrice Guez

A.I. Square Connect

Nicolas Paris

A.I. Square Connect

Date Written: October 2, 2018

Abstract

In this paper, we present three remarkable properties of the normal distribution: first that if two independent variables 's sum is normally distributed, then each random variable follows a normal distribution (which is referred to as the Levy Cramer theorem), second a variation of the Levy Cramer theorem that states that if two independent symmetric random variables with finite variance have their sum and their difference independent, then each random variable follows a standard normal distribution, and third that the normal distribution is characterized by the fact that it is the only distribution for which the sample mean and variance are independent, which is a central property for deriving the Student distribution and referred as the Geary theorem. The novelty of this paper is to provide new, quicker or self contained proofs of theses theorems.

Keywords: Geary Theorem, Levy Cramer Theorem, Independence Between Sample Mean and Variance

Suggested Citation

Benhamou, Eric and Guez, Beatrice and Paris, Nicolas, Three Remarkable Properties of the Normal Distribution (October 2, 2018). Available at SSRN: https://ssrn.com/abstract=3260190 or http://dx.doi.org/10.2139/ssrn.3260190

Eric Benhamou (Contact Author)

A.I. Square Connect ( email )

35 Boulevard d'Inkermann
Neuilly sur Seine, 92200
France

LAMSADE- Paris Dauphine University ( email )

Place du Marechal de Lattre de Tassigny
Pais, 75016
France

HOME PAGE: http://https://www.lamsade.dauphine.fr/

Beatrice Guez

A.I. Square Connect ( email )

35 Boulevard d'Inkermann
Neuilly sur Seine, 92200
France

Nicolas Paris

A.I. Square Connect ( email )

35 Boulevard d'Inkermann
Neuilly sur Seine, 92200
France

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