How to calculate $P(X<x | Y= \pm y)$

Question

Suppose that $X, Y$ are gaussian random variables with mean $\mu_x, \mu_y$ respectively, the same variance $\sigma^2$ and the covariance $cov(X,Y)= \rho \sigma^2$. It is easy to calculate $P(X<x | Y= y)$. We can just calculate the corresponding conditional density $f(x|y)$ first, and integrate it over the desired domain. But How do we calculate $P(X<x | Y= \pm y)$ given two possible values on the condition?

Answer 1

How about: $$P(X<x\mid Y=\pm y)=\frac{P(X<x\mid Y=y)f_Y(y)+P(X<x\mid Y=-y)f_Y(-y)}{f_Y(y)+f_Y(-y)}$$