Comparing two Standard Normal Correlated Variables Let's say I have two standard normal variables $X, Y$ (both with mean 0 and variance 1) with correlation $\rho &gt; 0$. Can I make any conclusions about their joint distribution?

For example, can I figure out $P(X &gt; c,Y &gt; c)$ for some $c \in \mathbb{R}$? The joint density of $X$ and $Y$ is no longer symmetrical (since they aren't independent) so I can't really do any geometric tricks. Is there a joint CDF or something else I'm missing?
Just consider the function graphically, and all should be clear:



which represents the joint distribution:

$$p(x,y) \propto e^{{\bf x}^t {\bf M} {\bf x}}$$

where ${\bf M} = \{ \{1, \rho\}, \{\rho, 1)\} \}.$

This is joint distribution with correlation $\rho = .5$ highlighted by the criterion $x&lt;.5$ and $y&lt;.5$.  Any moment, or correlation is just over the relevant region.   So if you want to calculate moments (for instance), then just integrate over the appropriate region.

The red is of course symmetrical, as the distribution dictates.
Just consider the function graphically, and all should be clear:



which represents the joint distribution:

$$p(x,y) \propto e^{{\bf x}^t {\bf M} {\bf x}}$$

where ${\bf M} = \{ \{1, \rho\}, \{\rho, 1)\} \}.$

This is joint distribution with correlation $\rho = .5$ highlighted by the criterion $x&lt;.5$ and $y&lt;.5$.  Any moment, or correlation is just over the relevant region.   So if you want to calculate moments (for instance), then just integrate over the appropriate region.

The red is of course symmetrical, as the distribution dictates.