Math 6010 • Linear Models

(20 points) Let \(Z = (Z_{1}, Z_{2}, Z_{3})^{\top} \sim \mathcal{N}(0, I_{3})\) and define the random vector
\[
X =
\begin{pmatrix}
X_{1} \\
X_{2}
\end{pmatrix}
=
\begin{pmatrix}
Z_{1} + Z_{2} \\
Z_{2} + Z_{3}
\end{pmatrix}.
\]
\begin{enumerate}
\item Compute the mean vector \(\mathbb{E}[X]\) and the covariance matrix \(\Sigma = \operatorname{Cov}[X]\).
\item Does \(X\) have a probability density function on \(\mathbb{R}^{2}\)? Briefly justify your answer.
\item Does the joint moment generating function satisfy
\[
M_{X}(t_{1}, t_{2}) = M_{X}(t_{1}, 0)M_{X}(0, t_{2})?
\]
Explain your reasoning.
\item Let \(Q = \|X\|^{2}\). Identify a symmetric matrix \(A\) such that \(Q = Z^{\top}AZ\) and compute \(\mathbb{E}[Q]\).
\end{enumerate}