Math 6010 • Linear Models

Let \((X_{1}, X_{2})\) be independent normal random variables with \(\mathbb{E}X_{1} = \mathbb{E}X_{2} = 0\) and \(\mathbb{E}X_{1}^{2} =\)
\(1, \mathbb{E}X_{2}^{2} = 4\). Define \(Y_{1} = X_{1} + X_{2}\) and \(Y_{2} = X_{1} + 2X_{2}\) and let \(\mathbf{Y} = (Y_{1}, Y_{2})^{\top}\).
\begin{enumerate}

\item
Compute the covariance matrix of \(\mathbf{Y}\).


\item
Find a matrix \(\mathbf{D}\) such that \(\mathbf{Y}^{\top}\mathbf{D}\mathbf{Y}\) has a chi-squared distribution.

\end{enumerate}