The mean vector and covariance matrix of the trivariate random variable \(\mathbf{X}\) is given by

\(\mathrm{E}(\mathbf{X})=\left[\begin{array}{c}-2 \\ 4 \\ 2\end{array}ight]\) and \(\boldsymbol{\operatorname { C o v }}(\mathbf{X})=\left[\begin{array}{ccc}10 & 2 & 1 \\ 2 & 5 & 0 \\ 1 & 0 & 1\end{array}ight]\)

The random variable \(\mathbf{Y}\) is defined by \(Y=\mathbf{c}^{\prime} \mathbf{X}\), where \(\mathbf{c}^{\prime}=\left[\begin{array}{lll}5 & 1 & 3\end{array}ight]\), and the bivariate random vector \(\mathbf{Z}\) is defined by \(\mathbf{Z}=\mathbf{A X}\), where \(\mathbf{A}=\left[\begin{array}{lll}1 & 1 & 1 \\ 2 & 3 & -4\end{array}ight]\).

(a) Define as many of the values \(\mathrm{E}\left(X_{i} X_{j}ight)\), for \(i\) and \(j \in\) \(\{1,2,3\}\), as you can.

(b) Define the correlation matrix for \(\mathbf{X}\)

(c) Define the mean and variance of \(Y\).

(d) Define the mean vector, covariance matrix, and correlation matrix of \(\mathbf{Z}\).