If $X_1$ is $N(0,1)$ and $X_2 = -X_1$ if $-1 \leq X_1 \leq 1$ or $X_2 = X_1$ otherwise, how do you prove that $X_1$ and $X_2$ do not have a bivariate normal distribution?
Attempt: Using the Jacobian method I proved that the distribution of $X_2$ is the standard normal distribution. To prove that $X_1$ and $X_2$ do not have a bivariate normal distribution there is a hint that says to consider the linear combination $X_1-X_2$ where $P(X_1-X_2) = P(|X_1|>1) = 0.3174$.