1. Let $X' = (X_1, X_2, X_3) \sim N_3(\mu, \Sigma)$ where $\mu' = (2, 1, 2)$ and $\Sigma = \begin{bmatrix} 2 & 1 & 1 \ 1 & 3 & 0 \ 1 & 0 & 1 \end{bmatrix}$
i. Write down the trivariate normal density of X.
ii. Let $Y_1 = 2X_1 - X_2$ and $Y_2 = X_1 + X_2 + X_3$. Find $Cov(Y_1, Y_2)$ and hence the correlation coefficient between $Y_1$ and $Y_2$.
iii. Determine the conditional distribution, mean and variance of $X_3$ and $X_1$ on $X_2$.