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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.
Yes - take $Y=-X$ if $X<0$ and $Y$ be "negative gaussian" independent of the value of $X$ if $X>0$. Then $X+2Y=-X>0$ for $X<0$ (i.e., $X+2Y$ is positive at least with probability 1/2); on the other hand, if $X>0$ and $Y<0$ they are conditionally independent, so that $X+2Y$ can also be positive with non-zero probability.
