2. For the random vector $N_p(\mu, \Sigma)$, derive the conditional distribution of $X$ given the partitioned
mean and covariance matrix shown in equation (1):
$\begin{aligned}
X = \begin{bmatrix} X_1\\X_2 \end{bmatrix}, \quad \mu = \begin{bmatrix} \mu_1\\\mu_2 \end{bmatrix}, \quad \Sigma = \begin{bmatrix} \Sigma_{11} & \Sigma_{12}\\ \Sigma_{21} & \Sigma_{22} \end{bmatrix} \end{aligned}$ (1)
Assuming $\Sigma_{12} = 0$, which means $X_1$ and $X_2$ are uncorrelated, determine the conditional
distribution of $X_1$ given $X_2$.
(Hint: $M_{X_1, X_2}(s_1, s_2) := \mathbb{E}(e^{s_1 X_1 + s_2 X_2}) = M_{X_1}(s_1) \cdot M_{X_2}(s_2)$ if and only if $X_1$ and $X_2$ are
independent.)
