[gbrown]

Wouldn’t it be even faster to solve that problem conditionally? P(x >= a)P(y >= a | x >= a), since the conditional distribution is known and normal?

[mathrando]

You still need to integrate over x. y|x=c is Gaussian for each c, but there are many such c's in [a,inf].

If you're curious:

https://www.ams.org/journals/mcom/1978-32-141/S0025-5718-197...

[gbrown]

Ah, duh. Not enough coffee.