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?

What you need is $\mathbb{E}(X_1 \mid Y_1, Y_2)$.  We have
$$
\operatorname{var}\begin{bmatrix} X_1 \\  Y_1 \\  Y_2 \end{bmatrix} = \left[\begin{array}{r|rr} 1 & 1 & -0.9 \\  \hline1 & 1.02 & -0.9 \\ -0.9 & -0.9 & 1.02 \end{array}\right]= \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{12}^\top & \Sigma_{22} \end{bmatrix}.
$$
So the conditional expected value is
$$
\mathbb{E}(X_1) + \Sigma_{12} \Sigma_{22}^{-1}
\left(
\begin{bmatrix} Y_1 \\ Y_2 \end{bmatrix} - \mathbb{E}\begin{bmatrix} Y_1 \\ Y_2 \end{bmatrix}.
\right)
$$
See: http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Conditional_distributions
