Consider two standard normal random variable, X and Y.

They both have mean 0, and variance 1. But we don't know their dependency. Is it possible for X+2Y to be nonsymmetric?

In another word, is it possible for P(X+2Y>0) = 1/2 to not hold.

I understand if they follow multivariate normal, the sum has to be normal and thus symmetric.

Is it possible to construct non-jointly Gaussians X and Y such that the equality does not hold.