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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 $\rho$ .

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    $\begingroup$ Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. $\endgroup$ – José Carlos Santos Jul 12 at 8:50
  • $\begingroup$ Might this depend on $\rho$ ? $\endgroup$ – Henry Jul 12 at 15:53
  • $\begingroup$ maybe. i dont know the answer $\endgroup$ – alexprice Jul 14 at 14:51

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