What is $P(X+Y>0 \mid X<0)$ given $X \sim N(0,\sigma^2)$, $Y\sim N(0,\sigma^2)$, $Corr(X,Y)=\rho>0$?

Question

I am trying to solve the problem indicated in the title. I found a similar one in this link, What is $P(X+Y>0 \mid X>0)$ given that $X,Y$ two different normal?

My problem differs from the linked one as $X$ and $Y$ are correlated in my problem. But I un unclear about from which step should the solution to my problem differs from the linked one. I guess $\rho$ should show up in the last step. Could someone help? Thanks!

Answer 1

Hint:

(1) the joint density function of $X,Y$ is given as $$f_{X,Y}(x,y)= \frac{1}{2\pi\sigma^2\sqrt{(1-\rho^2)}}\exp{\left(-\frac{1}{2\sigma^2(1-\rho^2)}(x^2-2\rho xy+y^2)\right)}$$

(2) by the definition of conditional probability, we have $$P(X+Y>0|X<0)=\frac{P(X+Y>0,X<0)}{P(X<0)}$$ The denominator is $\frac12$ by symmetry, the numerator can be evaluated as a double integral on the set $\left\{(x,y)\in \mathbb R^2:x+y>0,x<0\right\}$.