Expectation of a normal random variable when conditioning on a correlated normal random variable being above a threshold Suppose $X$ and $Y$ are correlated with correlation coefficient $\rho$. They are jointly normal with means $\mu_X$ and $\mu_Y$ respectively. Then what is $E[X | Y \geq T]$? Feel free to add additional assumptions if necessary.
In the case where $X = Y$, we can use the formula here: https://en.wikipedia.org/wiki/Truncated_normal_distribution#One_sided_truncation_(of_lower_tail)[4].

As a closed form solution has already been proposed, here is a relatively simple numerical algorithm to arrive at an answer as well.

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*Generate a random uniform deviate between 0 and 1.


*Use this value in a software supplied Normal random deviate inversion algorithm to create a Normal deviate, say $Y_i$, for a Normal distribution with associated mean = $μ_y$ and sigma = $σ_y$.


*If $Y_i$ ≥ t, keep the sample point, otherwise return to Step 1.


*Compute $E[X|Y = Y_i ]$ per the well known formula $\mathbb E[X | Y_i ] = \mu_x + ρ\frac{\sigma_x}{\sigma_y}(Y_i -\mu_y) $. Source see, for example, this education reference.


*Repeat starting at Step 1 for a large number of iterations.


*Finally, calculate the average of all values obtained in Step 4.
Note, one can easily adjust Step 3 in this Monte Carlo simulation approach to handle more complex cases (like a range).
