1. Given a Gaussian random vector $X \sim N(\mu, \Sigma)$ where $\mu = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}^T$ and $\Sigma = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 4 & 1 \\ 0 & 1 & 2 \end{bmatrix}$.
a. What is the pdf of $Z = X_1 - X_2 - X_3$.
b. What is the probability $P(Z < 0)$ 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 $Y = BX + c$, where $B = \begin{bmatrix} 1 & -1 & 1 \\ 1 & 0 & 1 \end{bmatrix}$ and $c = \begin{bmatrix} -1 & -1 \end{bmatrix}^T$.