  We study when a given Gaussian random variable on a given probability space
$(\Omega, {\cal{F}}, P) $ is equal almost surely to $\beta_{1}$ where $\beta $
is a Brownian motion defined on the same (or possibly extended) probability
space. As a consequences of this result, we prove that the distribution of a
random variable (satisfying in addition a certain property) in a finite sum of
Wiener chaoses cannot be normal. This result also allows to understand better
some characterization of the Gaussian variables obtained via Malliavin
calculus.
