I would like to find the conditional bivariate normal distribution. 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 $P(X|Y>M)$.
I found the conditional expectation of $X$ given that $Y$ is bigger than $M$: $E(X|Y>M)= \mu + \rho \sigma \frac{\phi(\frac{M-\mu}{\sigma})}{1-\Phi(\frac{M-\mu}{\sigma})}$.
But what is the conditional variance of $var(X|Y>M)$? Is it $(1-\rho^2)\sigma^2 $, as it would be in the case of $var(X|Y=M)$, where variance does not depends on $M$?
And is the conditional distribution $N(E(X|Y>M),var(X|Y>M))$?