Conditional distribution of two Gaussians given one is larger.

Question

Given two Gaussian random variables $Z:=(Z_1,Z_2)\sim N(0,\Sigma)$, where $\Sigma\in\mathbb{R}^{2\times 2}$, is there an explicit distribution for $Z$ conditioned on $Z_1>Z_2$? Is it still Gaussian? I know $\mathbb{P}[Z_1>Z_2]$ has an explicit expression which led me to believe maybe $(Z|Z_1>Z_2)$ has an explicit expression too.

For example, using $\Sigma=[[1, 0.5],[0.5, 1]]$ and plotting the histograms of the conditioned $Z_1$ and $Z_2$, they look Gaussian to me.

Picture of $Z_1|Z_1>Z_2$:

Picture of $Z_2|Z_1>Z_2$:

Answer 1

$Z = (Z_1,Z_2)$ has a bivariate Gaussian density whose support is the entire plane. Conditioned on $Z_1>Z_2$, the conditional density of $Z$ has support only a half-plane and so the conditional density of $Z$ is very definitely not bivariate Gaussian. But it is entirely possible for two marginally Gaussian random variables to have joint density function that is not a bivariate Gaussian density. Indeed, as user cardinal over on stats.SE remarks

The bivariate normal distribution is the exception, not the rule!

when two random variables are marginally Gaussian (normal) and he provides numerous examples to back up his assertion.