There are two dependent normal variables with the same distribution and the correlation coefficient $\rho$: $X,Y \sim N(\mu, \sigma^2)$.

I would like to get $P(X>M|Y>M)$. For that I need to know the conditional expectation and variance. But I can not find any information about the conditional normal variance in the case when $Y$ is smaller/bigger than some real number.

There is not a comparison between two normal variables. And it's not the truncated normal distribution.

I found the conditional expectation of $X$ given that $Y$ is bigger than $M$ is $E(X|Y>M)= \mu + \rho \sigma \frac{\phi(\frac{M-\mu}{\sigma})}{1-\Phi(\frac{M-\mu}{\sigma})}$.

But what is the conditional variance $var(X|Y>M)$? Is it $(1-\rho^2)\sigma^2 $, as it would be in the case of $var(X|Y=M)$, where variance does not depend on $M$?