Suppose I have a 3D multivariate gaussian, $[X_1,X_2,X_3]$. $var(X_i) = 1, cov(X_2,X_3)=cov(X_1,X_2)=0.98, \mu=0$. As I vary $cov(X_1,X_3)$ from $0.921$ (~ the minimum so that the covariance matrix is positive semidefinite) to ~ 1 (can get as close as we want), a thing happens that is confusing me, and I'm hoping someone here can shed some light.
Suppose that $X_1$ is given as -1 and $X_2$ as 1. The mean/median/mode value of $X_3$ is very different depending on the covariance. As $cov(X_1,X_3)$ goes to 1, the mean/median/mode of $X_3$ ~ -1. This is expected, since $X_1$ was measured at -1, and $X_3$ covaries with $X_1$ more strongly than with $X_2$. However, as $cov(X_1,X_3)$ goes to ~0.921, the mean/median/mode of $X_3$ goes to 3!
I would expect that $X_3$ would be near $X_2$=1, or, at the very least, somewhere between $X_1$=-1, $X_2$=1 or the mean of 0. What is the intuition behind this 3?