  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.
