Gaussian correlation inequality

The Gaussian correlation inequality is a result concerning the behavior of Gaussian random variables and their correlations. Specifically, it states that if \( X_1 \) and \( X_2 \) are two jointly distributed Gaussian random variables with the same variance, then their correlation satisfies a specific property regarding their joint distribution. Formally, if \( X_1 \) and \( X_2 \) are standard normal random variables (i.e.