Given a Gaussian random vector $xN(,)$ where $={\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^T$ and $=\left[\begin{array}{ccc}1& 0& 1\\ 0& 9& 0\\ 1& 0& 4\end{array}\right].$

a$.$ What is the pdf of $Z=x_1+x_3.$

b$.$ What is the probability $P(Z<mn\; id="EditorElement\_12853358">0</mn><mi\; id="EditorElement\_12853360">)</mi>$ in terms of Q$-$functions.

c$.$ Find the MMSE estimate of $x_3$ given $(x_1,x_2).$

d$.$ What is the joint pdf of $Y=Ax+b,$ where $A=\left[\begin{array}{ccc}2& 0& 1\\ 0& -1& 0\end{array}\right]$ and $b={\left[\begin{array}{cc}-3& -1\end{array}\right]}^T.$