1
$\begingroup$

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.

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.