Data sampled at two time instances giving bivariate Gaussian vector $X=(X_1,X_2)^T$ with
Data measured in noisy environment with vector: $(Y_1,Y_2)^T=(X_1,X_2)^T+(W_1,W_2)^T$
where $W_1,W_2$ are both $i.i.d.$ with $\sim N (0,0.2)$.
I have found correlation coefficient of $X_1,X_2$, $\rho=-0.9$ and $X_1,X_2 \sim N(0,1)$
Question: How to design filter to obtain MMSE estimator of $X_1$ from $Y$ vector and calculate MSE of this estimator?