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Let $X$ and $Y$ be jointly normal random variables with mean zero, variances $\sigma_x^2$ and $\sigma_y^2$, correlation $\rho$.

Then, given a constant $\delta$:

$\mathrm{E}\left[x\ |\ y>\delta\right]=\mathrm{E}\left[\mathrm{E}_y[x\ |\ y]\ |\ y>\delta\right] = \frac{\rho\sigma_x}{\sigma_y}\mathrm{E}[y\ |\ y>\delta]$

It is not clear to me how we went from inner expectation to the expression with $\frac{\rho\sigma_x}{\sigma_y}$ in it and factored out $x$ from the expectation.

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The conditional expectation of $X$ given $Y=y$ for mean zero bivariate normal $(X,Y)$ is

$$ \frac{\rho\sigma_x}{\sigma_y}y, $$ see e.g. here. We therefore do not need to factor out $x$, it no longer shows up in the conditional expectation.

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