Given X,Y bivariate normal with correlation $\rho$ and $0$ means,and stddevs = $1$ , K>$0$ , which variance is smaller:
$Var(Y1_{X<=K})$ or $Var(Y1_{X>K})$ ?
context: $E(Y1_{X<=K})$ , which is equal to $-E(Y1_{X>K})$ arises in several quantitative finance applications like Stochastic Local Volatility. Intuitively it's preferable to calculate $E(Y1_{X<=K})$ because the probabilty distribution is centered on $0$ and therefore has higher mass there. but would like to confirm that variance of $E(Y1_{X<=K})$ is smaller. It potentially could depend on $\rho$ .