An inequality about 2-dimensional normal random variable with a negative correaltion Suppose $(X,Y)$~$N$($0$,$0$,$\sigma_1^2$,$\sigma_2^2$,$\rho$), where $\rho<0$. How to prove for any positive number $a,b$, $P[X≥a,Y≥b]≤P[X≥a]P[Y≥b]$?

Look at E. H. Lehmann's "Some Concepts of Dependence", _Ann. Math. Statist._ , Volume 37, Number 5 (1966), 1137-1153, which contains your result.