TITLE: Linearly independent random variables and independent random variables
QUESTION [6 upvotes]: Does one of these two assertions imply the other ?
(1) $X_1, X_2, ..., X_n$  are linearly independent random variables (i.e. $\lambda_1 X_1 + \lambda_2 X_2 + ... + \lambda_n X_n = 0$ => $\lambda_1 =\lambda_2=...=\lambda_n=0$)
and
(2) $X_1, X_2, ..., X_n$  are independent random variables (stochastically independent)
If not, is there some special cases for which one implication (1=>2 or 2=>1) is true (Gaussian law? etc.)

REPLY [2 votes]: Neither implies the other.
As a counterexample of $1 \Rightarrow 2$ take $X_1 = X,\ X_2 = X^2$ for some r.v. $X$, say $X\sim U(0,1)$. They are of course not stochastically independent, but should be linearly independent.
For $2 \Rightarrow 1$, just use a.s. constant $X_1, X_2$. The are stochastically independent, but not linearly independent.