  In this paper, we compare two variances of maxima of $N$ standard Gaussian
random variables. One is a sequence of $N$ i.i.d. standard Gaussians, and the
other one is $N$ standard Gaussians with covariances $\sigma_{1,2}=\rho
\in(0,1)$ and $\sigma_{i,j}=0$, for other $ i\neq j$. It turns out that we need
to discuss the covariance of two functions with respect to multivariate
Gaussian distributions. Gaussian correlation inequalities hold for many
symmetric (with respect to the origin) cases. However, in our case, the max
function and its derivatives are not symmetric about the origin. We have two
main results in this paper. First, we prove a specific case for a
convex/log-concave correlation inequality for the standard multivariate
Gaussian distribution. The other result is that the variance of maxima of
standard Gaussians with $\sigma_{1,2}=\rho \in(0,1)$, while $\sigma_{i,j}=0$,
for other $ i\neq j$, is larger than the variance of maxima of independent
standard Gaussians. This implies that the variance of maxima of $N$ i.i.d.
standard Gaussians is decreasing in $N$.

Probability