Filter to obtain MMSE of data from Gaussian vector Data sampled at two time instances giving bivariate Gaussian vector $X=(X_1,X_2)^T$ with
$f(x_1,x_2)=\exp(-(x_1^2+1.8x_1x_2+x_2^2)/0.38)/2\pi \sqrt{0.19}$
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?

This looks like homework but here goes.  Since everything is Gaussian, the MMSE estimator for $X_1$ is the mean of the conditional pdf of $X_1$ given $(Y_1, Y_2)$ and the mean square error is the conditional variance of this.  Do you know how to find the conditional pdf (hint: it is also Gaussian)
