Three remarkable properties of the Normal distribution

TL;DR

This paper discusses three key properties of the normal distribution, providing new, quicker proofs for the Levy Cramer theorem, a variation involving symmetric variables, and the characterization by independence of sample mean and variance.

Contribution

It offers novel, more efficient proofs of three fundamental theorems characterizing the normal distribution.

Findings

Proofs of the Levy Cramer theorem are simplified.

A variation involving symmetric variables is established.

The normal distribution is uniquely characterized by the independence of sample mean and variance.

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.