Is it impossible that $(X,Y)$ follows bivariate normal if they always have the same sign?

Question

Suppose $X$ and $Y$ follows normal distribution respectively. I heard that if $X$ and $Y$ always have the same sign, then $(X,Y)$ cannot follow the bivariate normal distribution. However, I can't see why. Any explanation please?

Answer 1

A bivariate normal distribution, as commonly understood, is a joint distribution with a density function $f(x,y)$ that is strictly positive for all values of $x$ and $y$. The probability that $X$ and $Y$ have opposite sign is $P(X>0, Y<0) + P(X<0, Y>0)$, which will be positive since you are integrating a strictly positive joint density over the regions $(0,\infty)\times(-\infty,0)$ and $(-\infty,0)\times(0,\infty)$. So there is always a chance, albeit possibly tiny, that $X$ and $Y$ have opposite sign when $(X,Y)$ have bivariate normal distribution.

There is one situation where $X$ and $Y$ always have the same sign: when $X$ and $Y$ are equal to each other, i.e., $Y$ and $X$ are names for the same variable. But in that situation the joint distribution is "singular" and doesn't possess a density, and people regard this situation as a degenerate case.